%\documentstyle[epsf,12pt]{article}
\documentclass[12pt]{article}
\usepackage[dvips]{epsfig}
\usepackage{cite}
\hoffset -10mm
\voffset -10mm
\textwidth 165mm
\textheight 220mm
\newcommand{\ep}{\varepsilon}
\newcommand{\Gl}[2]{{\mbox{Gl}}_{#1}\left(#2\right)}
\newcommand{\Li}[2]{{\mbox{Li}}_{#1}\left(#2\right)}
\newcommand{\Cl}[2]{{\mbox{Cl}}_{#1}\left(#2\right)}
\newcommand{\Ls}[2]{{\mbox{Ls}}_{#1}\left(#2\right)}
\newcommand{\LS}[3]{{\mbox{Ls}}_{#1}^{(#2)}\left(#3\right)}                    
\newcommand{\Lsc}[2]{{\mbox{Lsc}}_{#1\!}\left(#2\right)}
\newcommand{\tfrac}[2]{{\textstyle{\frac{#1}{#2}}}}
\newcommand{\ST}[2]{\left[ #1 \atop #2\right]} 
\newcommand{\SN}[3]{{\mbox{S}}_{#1,#2}\left(#3\right)}
\newcommand{\HA}[2]{{\mbox{H}}^{(#1)}_{#2}}
\newcommand{\Snp}[2]{{\mbox{S}}_{#1\!}\left(#2\right)}
\newcommand{\lc}[1]{\ln \left( 2 \cos \tfrac{#1}{2} \right)}
\newcommand{\ls}[1]{\ln \left( 2 \sin \tfrac{#1}{2} \right)}
               
\def\H{\hbox{H}}
\def\d{\hbox{d}}
\def\f{\hbox{f}}

\renewcommand{\theequation}{\thesection.\arabic{equation}}
%=====================================================================
\begin{document}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\begin{flushright}
 {DESY/02-214} \\[3mm]
 {hep-th/0303162} \\[3mm]
 {March 2003}
\end{flushright}
 \vspace*{2.0cm}
 \begin{center}
 {\Large \bf
 Massive Feynman diagrams and inverse binomial sums
 }
 \end{center}
 
\vspace{3mm}
 
 \begin{center}
 A.~I.~Davydychev$^{a,}$\footnote{
 Current address: Schlumberger, SPC, 110 Schlumberger Dr., MD-5,
 TX~77478, USA.
 Email address:
 davyd@thep.physik.uni-mainz.de}
 \quad  and \quad
 M.~Yu.~Kalmykov$^{b,}$\footnote{
Supported by DFG under Contract SFB/TR~9-03. 
On leave from BLTP, JINR, 141980 Dubna, Russia.
Email address: kalmykov@ifh.de}\\
 
\vspace{3mm}

$^{a}${\em
Institute for Nuclear Physics, 
Moscow State University, \\
 119992, Moscow, Russia}
\\
\vspace{.3cm}
$^{b}${\em
 DESY--Zeuthen,
Theory Group, Platanenallee 6, \\ D-15738 Zeuthen, Germany}
\\
\end{center}
  
 \hspace{3in}
 \begin{abstract}
 When calculating higher terms of the $\ep$-expansion
 of massive Feynman diagrams, one needs to evaluate
 particular cases of multiple inverse binomial sums.
 These sums are related to the derivatives of certain
 hypergeometric functions with respect to their parameters.
 We explore this connection and
 analytically calculate a number of such infinite sums,
 for an arbitrary value of the argument which corresponds to
 an arbitrary value of the off-shell external momentum.
 In such a way, we find a number of new results for
 physically important two-loop two- and three-point
 Feynman diagrams. The results are presented in terms
 of generalized polylogarithmic functions.
\end{abstract}

\vspace{3mm}

%\noindent
%PACS: 11.15.Bt, 02.30.Gp, 02.30.Lt, 12.15.Lk
%%%% extra: 12.38.Bx 
%\\
%Keywords: Feynman diagrams, 
%          Hypergeometric functions, 
%          Multiple inverse binomial sums, 
%          Higher order $\ep$-expansion,
          
\thispagestyle{empty} 
\newpage
%=====================================================================
\renewcommand{\thefootnote}{\arabic{footnote}}
\setcounter{footnote}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In many cases,
the results of analytical calculation of Feynman diagrams can be 
represented as combinations of hypergeometric functions. 
However, the problem of constructing the $\ep$-expansion of
hypergeometric functions within dimensional regularization~\cite{dimreg} 
(where $n=4-2\ep$ is the space-time dimension) is not completely solved.
Rather often,  
results for the terms of the $\ep$-expansion can be expressed in terms of 
polylogarithms~\cite{Lewin} as well as 
Nielsen polylogarithms~\cite{Nielsen}. 
Recently it was demonstrated that new types of functions, 
harmonic polylogarithms~\cite{RV00} and 
multiple polylogarithms~\cite{GR2001}, appear in 
multiloop calculations. Furthermore, the nested sums~\cite{nested}
were proposed as a generalization of multiple polylogarithms, and 
the $\ep$-expansion for a large class of hypergeometric function 
was constructed. 

In this paper we study the {\it multiple inverse binomial sums}
defined as\footnote{Some particular results for 
the {\it multiple binomial sums} are presented in \cite{poleII}. 
Those sums are defined similarly to Eq.~(\ref{binsum}),
but with $\left( 2j \atop j\right)$ in the numerator, rather
than in the denominator.}
\begin{equation}
\Sigma_{a_1,\ldots,a_p;
\; b_1,\ldots,b_q;c}^{\; i_1,\ldots,i_p; \;j_1,\ldots,j_q}(u)
\equiv
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^c}
[S_{a_1}(j\!-\!1)]^{i_1}\ldots [S_{a_p}(j\!-\!1)]^{i_p}\;
[S_{b_1}(2j\!-\!1)]^{j_1}\ldots [S_{b_q}(2j\!-\!1)]^{j_q},
\label{binsum}
\end{equation}
where $S_a(j)\equiv\sum_{k=1}^j k^{-a}$ is the harmonic sum\footnote{
Through the rest of this paper, the notations $S_a$ and $\bar{S}_a$
will always mean $S_a(j-1)$ and $S_a(2j-1)$, respectively, 
even we do not mention this explicitly.} 
and $u$ is an arbitrary argument.
In what follows, we will also use
$z\equiv \tfrac{1}{4}u$
as the argument of the occurring hypergeometric functions.
For sums of the type~(\ref{binsum}), the
{\em weight} ${\bf J}$
can be defined as 
${\bf J} = c + \sum_{k=1}^{p} a_k i_k + \sum_{k=1}^{q} b_k j_k$, 
whereas the {\em depth} can be associated with the sum 
$\sum_{k=1}^{p} i_k + \sum_{k=1}^{q} j_k$.

The sums~(\ref{binsum})
appear in the calculation of massive Feynman diagrams 
within several different approaches: for instance, as solutions of 
differential equations for Feynman amplitudes~\cite{dem}, or
through a na\"ive $\ep$-expansion of hypergeometric functions 
within Mellin--Barnes technique~\cite{BD-TMF}.
Physical applications include the one-, two- and three-loop 
massive Feynman diagrams with two massive cuts 
\cite{FKV98,DD,FKV99,DK-bastei,DK01,oneloop}. 
The case $u=1$ corresponds to the single-scale propagator-type 
diagrams~\cite{single}. 
Although there are many publications 
concerning harmonic series~\cite{harmonic},
only a limited number of results are available for the inverse binomial sums. 
Some particular results 
for $u=1,2,3$ can be extracted from~\cite{KV00,DK01}.
The sums with $u=1,3$ are expressible in terms 
of an ``odd'' basis~\cite{odd}, whereas the case $u=2$
corresponds to an ``even'' basis~\cite{DK01,euler}\footnote{The connection 
between ``sixth root of unity''~\cite{B99}
and ``odd''/``even'' bases was discussed in~\cite{DK01,review}.}. 
However, only for special type of sums the analytical 
results are available~\cite{KV00,sums,BBK},
\begin{equation}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^c}   = 
 - \sum_{i=0}^{c-2} \frac{(-2)^i}{i! (c-2-i)!} 
( \ln u )^{c-2-i} \LS{i+2}{1}{\theta} \;,
\end{equation}
where
$c \geq 2$,
\begin{equation}
\label{def_theta}
\theta \equiv 2\arcsin\left(\tfrac{1}{2}\sqrt{u}\right) 
= 2\arcsin\sqrt{z}\; , 
\end{equation}
%
\begin{equation}
\label{def_Ls}
\LS{j}{k}{\theta} =   - \int\limits_0^\theta {\rm d}\phi \;
   \phi^k \ln^{j-k-1} \left| 2\sin\frac{\phi}{2}\right| \, ,
\quad 
\Ls{j}{\theta} = \LS{j}{0}{\theta} \; 
\end{equation}
is the generalized log-sine function~\cite{Lewin}.
Some examples when such functions occur in the $\ep$-expansion
of Feynman diagrams can be found in Refs.~\cite{Ls_ex,odd,D-ep}.
Usually, the occurring angles~(\ref{def_theta}),
possess certain geometrical meaning~\cite{DD}. 

The main aim of the present publication is the analytical 
calculation of inverse binomial sums~(\ref{binsum}),
including some relevant examples of physically important 
Feynman diagrams.

The paper is organized as follows. 
In Section~2, employing the connection with the $\ep$-expansion
of hypergeometric functions,
we obtain analytical results for sums of the type~(\ref{binsum}),
valid for $u\leq 4$ ($z\leq 1$). Mainly the sums of
the {\em weights}~${\bf 3}$ and ${\bf 4}$ are considered.  
In Section~3 the analytical continuation 
to other values of $u$ is constructed. 
Section~4 contains some applications of our results related to 
the $\ep$-expansion of Feynman diagrams, 
mainly two-loop master integrals. In Appendix~A we briefly summarize 
the relevant properties of the harmonic polylogarithms 
of complex arguments and related functions.
In Appendix~B we show how certain identities between hypergeometric
functions can be used to establish relations between the corresponding sums.
Appendix~C contains a realistic example of a physical application
of the considered integrals. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Inverse binomial sums and hypergeometric functions}
\setcounter{equation}{0}

\subsection{The $\ep$-expansion of the $_2F_1$ function}

Here, our analysis is based on comparing
two representations of hypergeometric function whose parameters
depend on $\varepsilon$. 
One of them is the series representation in terms of the harmonic sums,
whereas the second one is the exact result in terms of the
functions related to the polylogarithms. 

In particular, we consider the $_{2}F_1$ 
hypergeometric function of a special type,
\begin{equation}
_{2}F_1\left(\begin{array}{c|}
1+a_1\ep, 1+a_2 \ep\\
\tfrac{3}{2} + b \ep  \end{array} ~z \right)
= \sum_{j=0}^{\infty} \frac{u^j}{j!}\;
\frac{(1+a_1\ep)_j (1+a_2\ep)_j (2+b\ep)_j}  
{(3+2b\ep)_{2j}} \; ,
\end{equation}
where $u=4z$, $(\alpha)_j\equiv\Gamma(\alpha+j)/\Gamma(\alpha)$ is
the Pochhammer symbol,
and we have used the duplication formula
$(2\beta)_{2j}=4^j(\beta)_j(\beta+\tfrac{1}{2})_j$.
To perform the $\ep$-expansion we use the well-known 
representation
\begin{equation}
(1+a\ep)_j = j! \; \exp\left[ -\sum_{k=1}^{\infty} \frac{(-a\ep)^k}{k}
S_k(j) \right] \; ,
\label{pochhamer}
\end{equation}
which yields
\begin{eqnarray}
&& \hspace*{-15mm}
_{2}F_1\left(\begin{array}{c|}
1+a_1\ep, 1+a_2 \ep\\
\tfrac{3}{2} + b \ep  \end{array} ~z \right)
= \frac{2(1 + 2 b\ep)}{u} 
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) }  \frac{u^j}{j}  
\Biggl\{
1 
+ \ep \left[ 
   (A_1+b)S_1 - 2 b \bar{S}_1
      \right] 
\nonumber \\ && 
+  \ep^2 \Bigl[ 
2 b^2 \left( \bar{S}_2 + \bar{S}_1^2 \right) 
-  2 b (A_1+b) S_1 \bar{S}_1 
- \tfrac{1}{2} (A_2+b^2) S_2
+ \tfrac{1}{2} (A_1+b)^2 S_1^2
\Bigr]
\nonumber \\ && 
+ \ep^3 \Bigl[ 
\tfrac{1}{6} (A_1+b) ^3 S_1^3 
- b (A_1+b)^2 S_1^2 \bar{S}_1 
- \tfrac{1}{2} (A_1+b)(A_2+b^2)S_1 S_2
\nonumber \\ && \hspace{10mm}
+ 2 b^2 (A_1+b) S_1 \left( \bar{S}_2 + \bar{S}_1^2 \right)
+ b (A_2+b^2) S_2 \bar{S}_1
+ \tfrac{1}{6} (2 b^3 - A_1^3 + 3 A_1 A_2) S_3
\nonumber \\ && \hspace{10mm}
- \tfrac{4}{3} b^3 
\left( 2 \bar{S}_3 + 3 \bar{S}_2 \bar{S}_1 + \bar{S}_1^3 \right)
\Bigr]
+ {\cal O} (\ep^4)
\Biggr\} \; ,
\label{hyper:f21}
\end{eqnarray}
where $A_k \equiv a_1^k+a_2^k.$
Note that the coefficient of the $\ep^3$ term can be represented as
\begin{eqnarray}
\label{combinations}
&& \hspace*{-7mm}
\tfrac{1}{6}(A_1^3+b^3) C_0 
+2 b^2 (A_1+b) C_1
+4 b^3 C_2
+\tfrac{1}{2} b (A_1-b) (A_1-2b) C_3
\nonumber \\ &&
+\tfrac{1}{2} (A_1^2-A_2) (A_1+b) \left(S_1 S_2-S_3\right)
+\tfrac{1}{2} b (A_1^2-A_2) \left(S_3-2 S_2 \bar{S}_1\right) \; ,
\end{eqnarray}
where we have introduced the following combinations of 
harmonic sums:
\begin{eqnarray}
\label{combinations2}
C_0&=& S_1^3-3S_1S_2+2S_3\; ,
\nonumber \\
C_1&=& 
S_1^3-S_1 S_2+S_1 (\bar{S}_1^2+\bar{S}_2)
-\tfrac{5}{2} S_1^2 \bar{S}_1+\tfrac{3}{2} S_2 \bar{S}_1 \; ,
\nonumber \\
C_2&=&  \tfrac{3}{4}S_1 S_2 - \tfrac{3}{4}S_1^3
+\tfrac{3}{2}S_1^2\bar{S}_1 - S_2\bar{S}_1 
-\tfrac{1}{3} \left( 2\bar{S}_3 + 3 \bar{S}_1\bar{S}_2
+\bar{S}_1^3\right) \; ,
\nonumber \\
C_3&=& S_1^3-2 S_1^2 \bar{S}_1 - S_1 S_2 + 2 S_2 \bar{S}_1 \; .
\end{eqnarray}
To completely define the $\ep^3$ order of the expansion
of $_2F_1$, we need results for six combinations
of sums, according to the number of independent
combinations of the parameters $A_i$ and $b$.

Let us first consider the case $0\leq u \leq 4 $ ($0\leq z\leq 1$).
In this region  the following parametrization 
can be used:  
$u = 4 \sin^2\tfrac{\theta}{2}$ ($z= \sin^2\tfrac{\theta}{2}$),
where $0 \leq \theta \leq \pi.$
In the rest of this paper, we will use the short-hand notation 
\begin{equation}
L_\theta \equiv
\ln \left( 2 \cos \tfrac{\theta}{2} \right) \; , \qquad  
l_\theta \equiv
\ln \left( 2 \sin \tfrac{\theta}{2} \right) \; .
\end{equation}

For a few special cases, the $\ep$-expansion of the 
$_2F_1$ functions is known. 
First of all, the following relation holds~\cite{PBM3}:
\begin{equation}
{}_{2}F_1 \left(\begin{array}{c|} \!\! 1\!+\!\ep, 1 \!-\! \ep\! \\
\frac{3}{2} \end{array} ~\sin^2 \tfrac{\theta}{2} \right)\! =
\frac{\sin \left( \ep \theta \right)}{\ep \sin(\theta)} \; .
\label{basis_I}
\end{equation}
Its expansion contains only the even powers of~$\ep$.
Using it, it is easy to get 
\begin{eqnarray}
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} S_2 
&=&  \tfrac{1}{6} \theta^3 
\tan\tfrac{\theta}{2} \; ,
\label{S2}
\\
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} 
\frac{u^j}{j} \left( S_2^2 - S_4 \right)
&=&  \tfrac{1}{60} \theta^5 
\tan\tfrac{\theta}{2} \; ,
\label{S4}
\end{eqnarray}
etc. Then, there are some cases~\cite{D-ep,DK01} when 
an arbitrary term of the $\ep$-expansion can be calculated
in terms of log-sine functions, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
_2F_1 \left(\begin{array}{c|} 1, 1+\ep  \\
\tfrac{3}{2} \end{array} ~\sin^2 \tfrac{\theta}{2} \right)
&=& \frac{\left( 2 \cos \tfrac{\theta}{2} \right)^{-2\ep}}{\sin \theta} 
\sum_{j=0}^{\infty} \frac{(2\ep)^j}{j!}
\left[ \Ls{j+1}{\pi-\theta} - \Ls{j+1}{\pi} \right] \; ,
\label{basis_II}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
_2F_1 \left(\begin{array}{c|} 1, 1+\ep  \\
\tfrac{3}{2}+\ep \end{array} ~\sin^2 \tfrac{\theta}{2} \right)
 &=& -\frac{2(1+2\ep)}{\sin\theta (\sin\tfrac{\theta}{2})^{2\ep}}
 \sum_{j=0}^{\infty} \frac{(2\ep)^j}{j!} \Ls{j+1}{\theta} \; ,
\\
_2F_1 \left(\begin{array}{c|} 1, 1+\ep  \\
\tfrac{3}{2}+\tfrac{1}{2}\ep \end{array} ~\sin^2 \tfrac{\theta}{2} \right)
 &=& -\frac{1+\ep}{(2\sin\theta)^{1+\ep}}
 \sum_{j=0}^{\infty} \frac{\ep^j}{j!} \Ls{j+1}{2\theta} \; .
\end{eqnarray}
Moreover, for a more general case an integral representation 
can be obtained~\cite{PBM3,DK01},
\begin{equation}
{}_{2}F_1 \left(\begin{array}{c|} 1, 1 \!+\! a \ep \\
\frac{3}{2}+b \ep \end{array} ~\sin^2 \tfrac{\theta}{2} \right) =
\frac{(1\!+\! 2 b \ep)\left( 2 \cos \tfrac{\theta}{2} \right)^{2b\ep-2a\ep}}
{ \sin \theta \left( 2 \sin \tfrac{\theta}{2} \right)^{2b\ep}}
\int\limits_0^\theta {\rm d}\phi
\left(2 \sin \tfrac{\phi}{2} \right)^{2b\ep} 
\left(2 \cos \tfrac{\phi}{2} \right)^{2a\ep-2b\ep} . 
\label{basis_III}
\end{equation}
Any order of the $\ep$-expansion of Eq.~(\ref{basis_III})
can be expressed in terms of 
the generalized ``log-sine-cosine'' function $\Lsc{i,j}{\theta}$
whose properties are listed in Appendix~B of Ref.~\cite{DK01},
\begin{equation}
\label{Lsc}
\Lsc{i,j}{\theta}=-\int\limits_0^{\theta}
{\rm d}\phi\; \ln^{i-1}\Bigl|2\sin\tfrac{\phi}{2}\Bigr| \;
\ln^{j-1}\Bigl|2\cos\tfrac{\phi}{2}\Bigr| \; .
\end{equation}
Up to the level $i+j=5$, only one independent function
appears, $\Lsc{2,3}{\theta}$, that cannot be expressed
in terms of ordinary log-sine integrals. 
Note that $\Lsc{2,3}{\pi-\theta}$ can be reduced 
to $\Lsc{2,3}{\theta}$ by using 
Eq.~(A.27) of Ref.~\cite{DK01}.

Using Eqs.~(\ref{basis_II})--(\ref{basis_III}), we obtain 
the following results for the weight-{\bf 2} and weight-{\bf 3}
sums, in addition to Eq.~(\ref{S2}): 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
&& 
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} S_1 
=  2 \tan\tfrac{\theta}{2}
\Bigl[\Ls{2}{\pi- \theta}  - \theta L_\theta \Bigr] \;,
\label{S1a}
\\ && 
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} \bar{S}_1
=  
\tan\tfrac{\theta}{2}
\Bigl[
2 \Ls{2}{\pi- \theta}
+  \Ls{2}{\theta}
+  \theta l_\theta
- 2 \theta L_\theta
\Bigr] \; ,
\label{S1bar}
\\ && 
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)}  \frac{u^j}{j} S_1^2 
=  4 \tan\tfrac{\theta}{2}
\Bigl[
\Ls{3}{\pi-\theta} - \Ls{3}{\pi}
- 2 \Ls{2}{\pi-\theta} L_\theta
+ \theta L^2_\theta
+ \tfrac{1}{24} \theta^3 
\Bigr] \;, 
\hspace*{7mm}
\label{S1S1}
\\ && 
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)}
\frac{u^j}{j} S_1 \bar{S}_1
=
\tan\tfrac{\theta}{2}  
\Bigl\{
  5  \left[ \Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right]
-    \Ls{3}{\theta}
+ \tfrac{1}{2} \Ls{3}{2\theta}
\nonumber \\ && \hspace*{5mm}
+ 2  \Ls{2}{\pi-\theta} l_\theta
- 8  \Ls{2}{\pi-\theta} L_\theta
- 2 \Ls{2}{\theta} L_\theta
- 2 \theta l_\theta  L_\theta 
+ 4 \theta  L_\theta^2 
+ \tfrac{1}{12} \theta^3
\Bigr\} \; ,
\label{S1S1bar}
\\ && 
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)}
\frac{u^j}{j} 
\left( \bar{S}_2 + \bar{S}_1^2 \right) 
=
\tan\tfrac{\theta}{2}  
\Bigl\{
  6  \left[ \Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right]
- 3  \Ls{3}{\theta}
\nonumber \\ && \hspace*{5mm}
+ 4  \Ls{2}{\pi-\theta} l_\theta
- 8  \Ls{2}{\pi-\theta} L_\theta
+ 2  \Ls{2}{\theta} l_\theta
+ \Ls{3}{2\theta}
- 4 \Ls{2}{\theta} L_\theta
\nonumber \\ && \hspace*{5mm}
+ \theta l^2_\theta
- 4 \theta  L_\theta l_\theta
+ 4 \theta  L^2_\theta 
+ \tfrac{1}{12} \theta^3
\Bigr\} \; ,
\label{S2S1bar}
\end{eqnarray}
where $\Ls{3}{\pi}=-\tfrac{1}{2}\pi\zeta_2$.
In fact, all $\Ls{j}{\pi}$ are expressible 
in terms of the $\zeta$-function~\cite{Lewin}.

For the weight {\bf 4}, the results for some combinations
of sums involved in~(\ref{combinations}) can also be expressed
in terms of log-sine functions, using 
Eqs.~(\ref{basis_II})--(\ref{basis_III}),
\begin{eqnarray}
&& \hspace*{-10mm} 
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} C_0
= 
8 \tan\tfrac{\theta}{2}
\Bigl\{
\Ls{4}{\pi-\theta} \!-\! \Ls{4}{\pi} 
\nonumber \\ && \hspace{15mm} 
\!-\! 3 \Bigl[ \Ls{3}{\pi \!-\! \theta} \!-\! \Ls{3}{\pi} \Bigr] L_\theta
\!+\! 3 \Ls{2}{\pi \!-\! \theta} L^2_\theta
\!-\! \theta L^3_\theta
\Bigr\} \; ,
\label{combination:1}
\\ && \hspace*{-10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} C_1 
= 
\tan\tfrac{\theta}{2}  
\Bigl\{
\tfrac{2}{3} \Ls{4}{\theta}
- \tfrac{1}{3} \Ls{4}{ 2\theta}
- \tfrac{14}{3} \left[ \Ls{4}{ \pi - \theta} - \Ls{4}{ \pi} \right]
\nonumber \\ && 
+ 14 \left[ \Ls{3}{ \pi - \theta} - \Ls{3}{\pi} \right] L_\theta
- 2 \Ls{3}{\theta} l_\theta
+ \Ls{3}{2 \theta} \left[ L_\theta + l_\theta \right]
- 14 \Ls{2}{\pi-\theta} L_\theta^2
\nonumber \\ && 
+ 2 \Ls{2}{\pi-\theta} l_\theta^2
- 2 \Ls{2}{2 \theta} L_\theta l_\theta   
- \Ls{2}{2 \theta} L_\theta^2
- 2 \theta L_\theta l_\theta^2
- 2 \theta L_\theta^2 l_\theta
+ 4 \theta L_\theta^3 
\Bigr\} \;, 
\\ && \hspace*{-10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} C_2 
= 
\tan\tfrac{\theta}{2}  
\Bigl\{
\tfrac{1}{3} \Ls{4}{2 \theta}
- \Ls{4}{\theta}
+ 4 \left[ \Ls{4}{\pi - \theta} - \Ls{4}{\pi}  \right] 
- \Ls{3}{2 \theta}\left( l_\theta + L_\theta  \right) 
\nonumber \\ &&
- 12 \left[\Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right] L_\theta
+ 3 \Ls{3}{\theta} l_\theta 
+ 12 \Ls{2}{\pi-\theta} L_\theta^2  
- 3 \Ls{2}{\pi-\theta} l_\theta^2
\nonumber \\ &&
- \tfrac{1}{2} \Ls{2}{2\theta} l_\theta^2 
+ \Ls{2}{2\theta} L_\theta^2 
+ 2 \Ls{2}{2\theta} l_\theta L_\theta 
+ 2 \theta  l_\theta^2 L_\theta
+ 2 \theta  l_\theta L_\theta^2
- \tfrac{1}{3} \theta  l_\theta^3
- \tfrac{10}{3} \theta L_\theta^3 
\Bigr\} \; ,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\Ls{4}{\pi}=\tfrac{3}{2}\pi\zeta_3$
and the combination of the harmonic sums $C_j$ are defined in (\ref{combinations2}).
Finally, using (\ref{basis_III}) the result for the sum involving $C_3$ 
can be expressed in terms of the ${\mbox{Lsc}}$-function (\ref{Lsc}),
\begin{eqnarray}
\label{last4}
&& \hspace*{-10mm}
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} C_3
= 
- 4\tan\tfrac{\theta}{2}  
\Bigl\{ 
  2 \Lsc{2,3}{\theta}
+ 2  \left[ \Ls{4}{\pi-\theta} - \Ls{4}{\pi} \right]
\nonumber \\ && 
+ 2 \left[ \Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right] l_\theta
- 8 \left[ \Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right] L_\theta
- \Ls{3}{2 \theta} L_\theta 
+ 2 \Ls{3}{\theta} L_\theta 
\nonumber \\ && 
+ 8 \Ls{2}{\pi-\theta} L_\theta^2 
- 4 \Ls{2}{\pi-\theta} L_\theta  l_\theta   
+ \Ls{2}{2 \theta} L_\theta^2
- 2 \theta  L_\theta^3
+ 2 \theta   L_\theta^2 l_\theta
\Bigr\}  \; .
\end{eqnarray}
Some of the results (\ref{S1a})--(\ref{last4}) can be written 
in a slightly different form by means of relations~\cite{Lewin}
$$
\Ls{2}{\theta} = \Cl{2}{\theta} \; , \quad 
\Cl{2n}{\pi+\theta} = - \Cl{2n}{\pi-\theta} \; , \quad
\Cl{2n+1}{\pi+\theta} = \Cl{2n+1}{\pi-\theta} \; . 
$$

However, in this way one cannot obtain
results for the two remaining combinations 
in~(\ref{combinations}),
involving $A_1^2-A_2=2a_1a_2$,
since in Eq.~(\ref{basis_III})
we always have $a_1a_2=0$. An interesting relation
between these two functions is obtained in Appendix~B. 
It should also be noted that for higher hypergeometric
functions other combinations of sums may arise
at this level (see below).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Expansion of higher functions}

Let us consider 
the hypergeometric function of the following type:
\begin{equation}
\hspace*{-3mm}
_{P+1}F_P\left(\begin{array}{c|}
\tfrac{3}{2} + b_1 \ep,\ldots, \tfrac{3}{2} + b_{J-1} \ep, \; 
1+a_1\ep, \ldots, 1+a_K \ep, \; 2+d_1 \ep \ldots, 2+d_L \ep \\
\tfrac{3}{2} + f_1 \ep, \ldots, \tfrac{3}{2} + f_J \ep, \; 
1+e_1\ep, \ldots, 1+e_{R}\ep, \; 
2+c_1\ep, \ldots, 2+c_{K+L-R-2} \ep
\end{array} ~z \right),
\label{PFQ}
\end{equation}
where $P=K+L+J-2$.
Using the representation (\ref{pochhamer})
its $\ep$-expansion can be reduced to inverse binomial sums~(\ref{binsum}).
The original hypergeometric function (\ref{PFQ}) can be written as
(see details in Appendix~B of Ref.~\cite{DK01})
\begin{eqnarray}
&& \hspace*{-10mm}
_{P+1}F_P\left(\begin{array}{c|}
\{ \tfrac{3}{2} +b_i\ep\}^{J-1}, \; 
\{ 1+a_i\ep\}^K, \; \{ 2+d_i\ep\}^L  \\
\{ \tfrac{3}{2} + f_i\ep\}^J, \; 
\{ 1+e_i\ep \}^R,
\{ 2+c_i\ep \}^{K+L-R-2} 
\end{array} ~z \right)
\nonumber \\ && \hspace*{20mm}
= \frac{2}{u}
\frac{
\Pi_{s=1}^{K+L-R-2} (1+c_s\ep)  
\Pi_{k=1}^{J} (1+ 2 f_k\ep)}
{\Pi_{i=1}^{L} (1+d_i\ep)
\Pi_{r=1}^{J-1} (1 + 2 b_r\ep)}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) }  
\frac{u^j}{j^{K-R-1}}  
\Delta \;, 
\nonumber 
\end{eqnarray}
where $u=4z$,  
\begin{eqnarray}
\Delta &=& 
\exp \left[ \sum_{k=1}^{\infty} \frac{(-\ep)^k}{k}  
\left( S_k T_k + 2^k U_k \bar{S}_k + W_k j^{-k} \right) 
\right]
\nonumber \\
&=&  
1 
- \ep \left( 
   T_1 S_1 
+ W_1 j^{-1} 
+ 2 U_1 \bar{S}_1
\right)
+  \ep^2 \Bigl[ 
\tfrac{1}{2} j^{-2}\left( W_2 + W_1^2 \right) 
+ W_1 j^{-1} 
 \left( T_1 S_1 
+ 2 U_1 \bar{S}_1  \right)
\nonumber \\ &&
+  2 T_1 U_1 S_1 \bar{S}_1  
+ 2 U_2 \bar{S}_2 
+ 2 U_1^2 \bar{S}_1^2 
+ \tfrac{1}{2} T_2  S_2 
+ \tfrac{1}{2} T_1^2 S_1^2
\Bigr]
+ {\cal O}(\ep^3)\; , 
\label{expansion}
\end{eqnarray}
and we introduced the constants 
\begin{eqnarray*}
&& \hspace*{-8mm}
A_k \equiv \sum_{i=1}^{K} a_i^k, \quad 
B_k \equiv \sum_{i=1}^{J-1} b_i^k, \quad 
C_k \equiv \sum_{i=1}^{K+L-R-2} c_i^k, \quad 
D_k \equiv \sum_{i=1}^{L} d_i^k, \quad 
E_k \equiv \sum_{i=1}^{R} e_i^k, \quad
F_k \equiv \sum_{i=1}^{J} f_i^k, 
\nonumber \\
&& \hspace*{-8mm}
T_k \equiv B_k + C_k + E_k - A_k - D_k - F_k, \quad
U_k \equiv F_k - B_k, \quad
W_k \equiv C_k - D_k. 
\end{eqnarray*}
Some results for separate terms 
of expansion~(\ref{expansion}) are collected in Section~2.1. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the sums of the type~(\ref{binsum}) with an arbitrary integer power 
$c\geq 2$  and  $0 \leq u \leq 4$, the following one-fold integral 
representation \cite{BBK} is useful:
\begin{eqnarray}
\sum_{j=1}^\infty  \frac{u^j}{j^c} \frac{f(j)}{\left( 2j\atop j\right)} =
\frac{1}{(c-2)!} \int\limits_0^{\theta}
{\rm d}\phi \frac{\cos \tfrac{\phi}{2}}{\sin\tfrac{\phi}{2}}
\Bigl[\ln{u} - 2\ln \left(2 \sin \tfrac{\phi}{2} \right)\Bigr]^{c-2} 
\sum_{j=1}^\infty \frac{\left( 4 \sin^2\tfrac{\phi}{2} \right)^{j}}{j} 
\frac{f(j)}{\left( 2j\atop j\right)}  \; ,
\label{higher_c}
\end{eqnarray}
where $f(j)$ stands for an arbitrary combination of the harmonic sums.
In this manner, the original sum of the type~(\ref{binsum}) 
with an arbitrary positive integer parameter $c\geq 2$ is reduced
to a one-fold integral representation containing
a sum with $c=1$ in the integrand. 
Using the representation~(\ref{higher_c}) we can generalize our 
results~(\ref{S2}), (\ref{S4}), (\ref{S1a})--(\ref{last4})
to the case of an arbitrary integer $c>1$.
However, only for special types of sums these results are 
expressible in terms of generalized log-sine functions~(\ref{def_Ls}):
\begin{eqnarray}
&& \hspace*{-7mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{j=1}^\infty  \frac{u^j}{j^c} \frac{1}{\left( 2j\atop j\right)} S_2 =
- \tfrac{1}{6} \sum_{i=0}^{c-2} \frac{(-2)^i}{i!(c-2-i)!} 
\left(\ln u \right)^{c-2-i} \LS{i+4}{3}{\theta} \;, 
\label{S2G}
\\ && \hspace*{-7mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{j=1}^\infty  \frac{u^j}{j^c} \frac{1}{\left( 2j\atop j\right)}
\left( S_2^2-S_4 \right) =
- \tfrac{1}{60} \sum_{i=0}^{c-2} \frac{(-2)^i}{i!(c-2-i)!} 
\left(\ln u \right)^{c-2-i} \LS{i+6}{5}{\theta} \;, 
\label{S4G}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{eqnarray}
where $c \geq 2$, $0 \leq u \leq 4$, and the angle $\theta$ is
defined in Eq.~(\ref{def_theta}). 

Further results can be extracted 
from integral representation~(\ref{higher_c}) by using 
the integration rules for Clausen's function~\cite{Lewin} and 
the following relation:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\[
k \int\limits_0^\theta {\rm d}\phi\;
{\phi}^{k-1} \LS{j}{i}{m \phi}
=  \theta^k \LS{j}{i}{m\theta} - \frac{1}{m^k} \LS{j+k}{i+k}{m \theta}
\; . 
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this way, we obtain 
\begin{eqnarray}
\label{S1c2}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } 
\frac{u^j}{j^2} S_1   &=&  
  4 \Cl{3}{\pi - \theta} - 2 \theta \Cl{2}{\pi - \theta} + 3 \zeta_3 , 
\\ 
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } 
\frac{u^j}{j^2} \bar{S}_1
 &=&  -2 \Cl{3}{\theta} + 4  \Cl{3}{\pi-\theta}
- 2 \theta \Cl{2}{\pi- \theta}
-   \theta \Cl{2}{\theta}
+ 5 \zeta_3 \; , \hspace*{10mm}
\\ 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} 
\frac{u^j}{j^2} S_1^2 
&=&  
4 \theta \left[ \Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right]
-  4\left[ \Ls{2}{\pi-\theta} \right]^2
+ \tfrac{1}{24} \theta^4 \; ,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\  
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } 
      \frac{u^j}{j^2} S_1 \bar{S}_1
&=& 
 5 \theta \left[\Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right]
- \theta \Ls{3}{\theta} 
+ \tfrac{1}{2} \theta \Ls{3}{2 \theta}
\nonumber \\ && 
- 4 \left[\Ls{2}{\pi-\theta} \right]^2
+ \tfrac{1}{48} \theta^4 
- 2 \Ls{2}{\pi-\theta}  \Ls{2}{\theta} \; , 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\  
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } 
\frac{u^j}{j^2} 
\left( \bar{S}_1^2 + \bar{S}_2 \right)
&=& 
  6 \theta \left[\Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right]
- 3 \theta \Ls{3}{\theta}
+  \theta \Ls{3}{2 \theta}
\nonumber \\ &&
- 4 \left[\Ls{2}{\pi-\theta} \right]^2
- \left[\Ls{2}{\theta} \right]^2
+ \tfrac{1}{48} \theta^4
- 4 \Ls{2}{\pi-\theta}  \Ls{2}{\theta} \; .
\label{S2bar_2}
\end{eqnarray}

However, other sums are not expressible in terms of 
the generalized log-sine functions.
One needs to introduce a new function\footnote{Alternatively,
instead of $\Phi(\theta)$ one may introduce the generalized
Glashier function $\Gl{4}{\theta;1}$, see Eq.~(\ref{Gl4(theta,1)}) 
in Appendix~A.},
\begin{equation}
\label{def_Phi}
\Phi(\theta) \equiv
\int\limits_0^\theta {\rm d}\phi\; 
\Ls{2}{\phi} \ln \left(2 \cos \tfrac{\phi}{2} \right) \; , 
\end{equation}
which obeys the following symmetry property:
\begin{equation}
\label{sym_Phi}
\Phi(\theta) + \Phi(\pi-\theta) 
= \Phi(\pi) + \Ls{2}{\pi-\theta} \Ls{2}{\theta} \;, 
\end{equation}
where 
\begin{eqnarray}
\hspace{-10mm}
\Phi(\pi) & = &  
\tfrac{1}{6} \ln^4 2 - \zeta_2 \ln^2 2 + \tfrac{7}{2} \zeta_3 \ln 2 
- \tfrac{53}{16} \zeta_4 + 4 \Li{4}{\tfrac{1}{2}} 
= 0.64909 \ldots 
\; . 
\end{eqnarray}

The following sums of the weight {\bf 4} are expressible 
in terms of the function $\Phi(\theta)$:
\begin{eqnarray}
&& \hspace*{-10mm}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } 
\frac{u^j}{j^3} S_1   
=  
  2  \left[  \LS{4}{1}{\pi-\theta} - \LS{4}{1}{\pi} \right]  
- \tfrac{1}{2} \LS{4}{1}{2 \theta}
+ 2 \LS{4}{1}{\theta}
- 4 \theta \Ls{2}{\pi-\theta} l_\theta
\nonumber \\ && 
- 2 \pi \left[\Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right]
+ 8 \left[ \Cl{3}{\pi-\theta} - \Cl{3}{\pi} \right] l_\theta
+ 4 \Phi(\pi-\theta) - 4 \Phi(\pi) \;, \hspace*{6mm}
\label{S1_3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\ && \hspace*{-10mm} 
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } 
\frac{u^j}{j^3} \bar{S}_1   
=  
  2 \left[ \LS{4}{1}{\pi-\theta} - \LS{4}{1}{\pi} \right]
- \tfrac{1}{2} \LS{4}{1}{2 \theta}
+ 4 \LS{4}{1}{\theta}
+ \left[ \Ls{2}{\theta} \right]^2
\nonumber \\ && 
- 2 \pi \left[\Ls{3}{\pi-\theta} - \Ls{3}{\pi} \right]
+ 8 \left[ \Cl{3}{\pi-\theta} - \Cl{3}{\pi} \right] l_\theta
- 4 \left[ \Cl{3}{\theta} - \zeta_3 \right] l_\theta
\nonumber \\ && 
- 2 \theta \Ls{2}{\theta} l_\theta
- 4 \theta \Ls{2}{\pi-\theta} l_\theta
+ 4 \Phi(\pi-\theta) - 4 \Phi(\pi) \;,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{S1bar_3}
\end{eqnarray}
%
where we have also used the following integral:
\begin{eqnarray}
\int\limits_0^\theta {\rm d}\phi\;
\phi \ln  \left(2 \cos \tfrac{\phi}{2} \right) 
     \ln  \left(2 \sin \tfrac{\phi}{2} \right)    
& =  & 
-\tfrac{1}{8} \LS{4}{1}{2\theta} + \tfrac{1}{2} \LS{4}{1}{\theta}
+ \tfrac{1}{2} \left[\LS{4}{1}{\pi - \theta} - \LS{4}{1}{\pi} \right]
\nonumber \\ && 
- \tfrac{1}{2}\pi \left[\Ls{3}{\pi - \theta} - \Ls{3}{\pi} \right] \; .
\nonumber 
\end{eqnarray}

Let us note that $\Phi(\theta)$ 
can be related to the real part of a certain
harmonic polylogarithm~\cite{RV00} of complex argument,
\begin{equation}
\label{PhiH}
\Phi(\theta) 
=  
\tfrac{1}{96} \theta^2 (2\pi-\theta)^2
- L_{\theta} \Cl{3}{\theta}
+ \zeta_3 \ln 2
-  \H_{-1,0,0,1}(1)   
+ \tfrac{1}{2}\Bigl[  
\H_{-1,0,0,1}( e^{{\rm i}\theta}) + \H_{-1,0,0,1}(e^{-{\rm i}\theta})   
\Bigr] ,
\end{equation}
where (for details, see Appendix~A) 
\begin{eqnarray}
H_{-1,0,0,1}(y) &=& \int\limits_0^y {\rm d}x \; \frac{\Li{3}{x}}{1+x} ,
\label{h(-1001)}
\\
\H_{-1,0,0,1}(1) &=&  
- \tfrac{1}{12} \ln^4 2 + \tfrac{1}{2} \zeta_2 \ln^2 2
- \tfrac{3}{4} \zeta_3 \ln 2 + \tfrac{3}{2} \zeta_4 - 2 \Li{4}{\tfrac{1}{2}}
=  0.33955 \ldots \; .
\end{eqnarray}
The function $\H_{-1,0,0,1}(y)$ has a branch cut starting 
at the point $y=-1$ ($\theta=\pm\pi$),
which is subtracted by logarithmic terms in~(\ref{PhiH}), so that 
$\Phi(\theta)$ is a smooth function of variable $\theta$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analytical continuation}
\setcounter{equation}{0}

To obtain results valid in other regions of variable $u$ (for
$u<0$ and $u>4$), we will construct
the proper analytical continuation of the expressions presented
in the previous section. 
For generalized log-sine integrals it is described in~\cite{DK01}.
Let us introduce a new variable
\begin{equation}
y \equiv e^{ {\rm i} \sigma \theta}, \hspace{5mm}
\ln(-y-{\rm i}\sigma 0) = \ln{y} - {\rm i} \sigma \pi,
\end{equation}
where the choice of the sign $\sigma=\pm 1$ is related to the
causal ``+i0'' prescription for the propagators\footnote{Whenever 
possible, we shall keep $\sigma$ undetermined, since one may need
different signs in different situations.}.
For completeness, we also present  
the inverse relations,
\begin{equation}
u = - \frac{(1-y)^2}{y} \;, \quad  
y = \frac{1-\sqrt{\frac{u}{u-4}}}{1+\sqrt{\frac{u}{u-4}}} \;, \quad
u \frac{{\rm d}}{{\rm d} u} 
= -\frac{1-y}{1+y} \; y \frac{{\rm d}}{{\rm d}y}  \; . 
\label{y<->u}
\end{equation}
In terms of this variable $y$, the analytic continuation of
all generalized log-sine integrals can be expressed in terms 
of Nielsen polylogarithms, whereas for
the function $\Phi(\theta)$ we get
\begin{eqnarray}
\Phi(\theta) & = & \zeta_3 \ln 2 + \tfrac{1}{2} \zeta_4 
-  H_{-1,0,0,1}(1) 
+ \tfrac{1}{2} \left[H_{-1,0,0,1}(y) + H_{-1,0,0,1}(y^{-1}) \right]
\nonumber \\ && 
- \tfrac{1}{4} \left[\Li{4}{y} + \Li{4}{y^{-1}} \right]
- \tfrac{1}{2} \left[ \ln(1+y) - \tfrac{1}{2} \ln y \right] 
           \left[\Li{3}{y} + \Li{3}{y^{-1}} \right] \;,
\end{eqnarray}
with $H_{-1,0,0,1}(y)$ 
defined in Eq.~(\ref{h(-1001)}).

Below we list the most complicated results, corresponding to the
analytical continuation of the results obtained in Section~2:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\label{AN_S1S1_2}
&& \hspace*{-10mm} 
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } \frac{u^j}{j^2} S_1^2
 =
- 8 \Snp{1,2}{-y} \ln y
+ 4 \Li{3}{-y} \ln y
- 2 \Li{2}{-y} \ln^2 y
+ 4 \left[ \Li{2}{-y} \right]^2
\nonumber \\ && \hspace{20mm}
- \tfrac{1}{24} \ln^4 y
+ 4  \zeta_2 \Li{2}{-y}
+ \zeta_2 \ln^2 y
+ 4 \zeta_3 \ln y
+ \tfrac{5}{2} \zeta_4 \; ,
\\ && \hspace*{-10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{AN_S1Sb1_2}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } \frac{u^j}{j^2} S_1 \bar{S}_1
 =  
- 10 \Snp{1,2}{-y} \ln y 
+  \Snp{1,2}{y^2} \ln y 
- 2 \Snp{1,2}{y} \ln y 
+ 4 \left[\Li{2}{-y} \right]^2
\nonumber \\ && 
- 2 \Li{2}{y} \Li{2}{-y} 
+ 3 \Li{3}{-y} \ln y 
- \Li{3}{y} \ln y 
- \tfrac{3}{2} \Li{2}{-y} \ln^2 y 
+ \tfrac{1}{2} \Li{2}{y} \ln^2 y 
\nonumber \\ && 
- \tfrac{1}{48} \ln^4 y 
+ 6 \zeta_2 \Li{2}{-y}
-  \zeta_2 \Li{2}{y}
+ \tfrac{5}{4} \zeta_2 \ln^2 y 
+ \tfrac{11}{2} \zeta_3 \ln y 
+ 5 \zeta_4 \; ,
\\ && \hspace*{-10mm} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{AN_Sb2_2}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } \frac{u^j}{j^2} 
\left( \bar{S}_1^2 +  \bar{S}_2 \right)
 =  
- 12 \Snp{1,2}{ -y} \ln y 
+ 2 \Snp{1,2}{y^2} \ln y 
- 6 \Snp{1,2}{y} \ln y 
+ 4 \left[\Li{2}{-y} \right]^2
\nonumber \\ && 
+  \left[\Li{2}{y} \right]^2
- 4 \Li{2}{-y} \Li{2}{y} 
+ 2 \Li{3}{-y} \ln y 
- \Li{3}{y} \ln y 
- \Li{2}{-y} \ln^2 y 
\nonumber \\ && 
+ \tfrac{1}{2} \Li{2}{y} \ln^2 y 
+ 8 \zeta_2 \Li{2}{-y}
- 4 \zeta_2 \Li{2}{y}
+ \zeta_2 \ln^2 y 
+ 8 \zeta_3 \ln y 
+ 10 \zeta_4 \; , 
\\ && \hspace*{-10mm} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } \frac{u^j}{j} C_0
 =  
\sigma \frac{1-y}{1+y}
\Bigl[
  24 \Snp{2,2}{-y}
- 48 \Snp{1,3}{-y}
- 12 \Li{4}{-y}
+ 24 \Li{3}{-y} \ln (1+y) 
\nonumber \\ && 
- 48 \Snp{1,2}{-y} \ln (1+y)
- 8 \ln y \ln^3 (1+y) 
+ 6 \ln^2 y \ln^2 (1+y) 
- 2 \ln^3 y \ln (1+y)
\nonumber \\ && 
+ \tfrac{1}{4} \ln^4 y 
+ 24 \zeta_3 \ln(1+y) 
- 12 \zeta_3 \ln y
- 24 \ln^2 (1+y) \Li{2}{-y}
+ 12 \zeta_2  \ln y \ln (1+y) 
\nonumber \\ && 
- 3 \zeta_2   \ln^2 y 
- 12 \zeta_2   \ln^2 (1+y) 
- \tfrac{27}{2} \zeta_4
\Bigr] \; ,
\\ && \hspace*{-10mm} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} C_1
= \sigma \frac{1-y}{1+y}
\Bigl[ 
  3 \Li{4}{-y}  
- 3 \Li{4}{y}
+ 28 \Snp{1,3}{-y} 
- 2  \Snp{1,3}{y^2} 
+ 4  \Snp{1,3}{y} 
\nonumber \\ &&
- 14 \Snp{2,2}{-y} 
+ \Snp{2,2}{y^2} 
- 2\Snp{2,2}{y}
+ 28 \Snp{1,2}{-y} \ln(1+y)
- 2 \Snp{1,2}{y^2} \ln(1-y)
\nonumber \\ && 
- 2 \Snp{1,2}{y^2} \ln(1+y)
+ 4 \Snp{1,2}{y} \ln(1-y)
+ 4 \Li{3}{-y} \ln (1-y) 
+ 2 \Li{3}{y} \ln (1-y) 
\nonumber \\ && 
- 10 \Li{3}{-y} \ln (1+y) 
+ 4 \Li{3}{y} \ln (1+y)  
- 2 \Li{2}{-y} \ln^2 (1-y) 
\nonumber \\ && 
- 2 \Li{2}{y^2} \ln (1+y) \ln (1-y) 
+ 12 \Li{2}{-y} \ln^2 (1+y) 
- 2 \Li{2}{y} \ln^2 (1+y) 
\nonumber \\ && 
- 2 \ln y \ln^2 (1-y) \ln (1+y) 
- 2 \ln y \ln (1-y) \ln^2 (1+y) 
+ 4 \ln y \ln^3 (1+y) 
\nonumber \\ && 
+ \tfrac{1}{2} \ln^2 y \ln^2 (1-y) 
+ 2 \ln^2 y \ln (1-y) \ln (1+y)  
- \tfrac{5}{2} \ln^2 y \ln^2 (1+y) 
- \tfrac{1}{2} \ln^3 y \ln (1-y) 
\nonumber \\ && 
+ \tfrac{1}{2} \ln^3 y \ln (1+y) 
+ \tfrac{45}{8} \zeta_4 
- \zeta_3 \ln (1-y) 
- 13 \zeta_3 \ln (1+y) 
+ 7 \zeta_3 \ln y 
+ \tfrac{9}{4} \zeta_2 \ln^2 y 
\nonumber \\ && 
- \zeta_2 \ln^2 (1-y)
+ 2 \zeta_2 \ln (1-y) \ln(1+y)
+ 8 \zeta_2 \ln^2 (1+y) 
- 9 \zeta_2 \ln y \ln (1+y) 
\Bigr] \;, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\ && \hspace*{-10mm}
\label{AN_last4}
\sum_{j=1}^\infty \frac{1}{\left( 2j\atop j\right)} \frac{u^j}{j} C_2
= \sigma \frac{1-y}{1+y}
\Bigl[
 2 \Snp{1,3}{y^2}
- 24 \Snp{1,3}{-y}  
- 6 \Snp{1,3}{y}
+ 12 \Snp{2,2}{-y} 
- \Snp{2,2}{y^2} 
\nonumber \\ && 
+ 3 \Snp{2,2}{y} 
- 2 \Li{4}{-y} 
+ \tfrac{5}{2} \Li{4}{y}
- 24 \Snp{1,2}{-y} \ln(1+y) 
- 6 \Snp{1,2}{y} \ln (1-y)  
\nonumber \\ && 
- \Li{3}{y} \ln (1-y) 
+ 2 \Snp{1,2}{y^2} \ln (1-y) 
+ 2 \Snp{1,2}{y^2} \ln (1+y) 
- 4 \Li{3}{-y} \ln (1-y) 
\nonumber \\ && 
+ 8 \Li{3}{-y} \ln (1+y) 
- 4 \Li{3}{y} \ln (1+y) 
+ 2 \Li{2}{-y} \ln^2 (1-y) 
- \Li{2}{y} \ln^2 (1-y) 
\nonumber \\ && 
+ 4 \Li{2}{-y} \ln (1-y) \ln (1+y) 
+ 4 \Li{2}{y} \ln (1-y) \ln (1+y) 
- 10 \Li{2}{-y} \ln^2 (1+y) 
\nonumber \\ && 
+ 2 \Li{2}{y} \ln^2 (1+y) 
- \tfrac{1}{3} \ln y \ln^3 (1-y) 
+ 2 \ln y \ln^2 (1-y) \ln (1+y) 
\nonumber \\ && 
+ 2 \ln y \ln (1-y) \ln^2 (1+y) 
- \tfrac{10}{3} \ln y \ln^3 (1+y) 
- \tfrac{1}{4} \ln^2 y \ln ^2 (1-y) 
\nonumber \\ && 
- 2 \ln^2 y \ln (1-y) \ln (1+y) 
+ 2 \ln^2 y \ln^2 (1+y) 
+ \tfrac{5}{12} \ln^3 y \ln (1-y) 
- \tfrac{1}{3} \ln^3 y \ln (1+y) 
\nonumber \\ && 
- \tfrac{1}{96} \ln^4 y 
- \tfrac{9}{4} \zeta_4 
+ 2 \zeta_3 \ln (1-y) 
+ 11 \zeta_3 \ln (1+y) 
- \tfrac{13}{2} \zeta_3 \ln y 
- 7 \zeta_2 \ln^2 (1+y) 
\nonumber \\ && 
+ 2 \zeta_2 \ln^2 (1-y) 
- 2 \zeta_2 \ln (1-y) \ln (1+y) 
- \tfrac{7}{4} \zeta_2 \ln^2 y 
- \zeta_2 \ln y \ln (1-y) 
\nonumber \\ && 
+ 8 \zeta_2 \ln y \ln (1+y) 
\Bigr] \;, 
\\ && \hspace*{-10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{AN_S1_3}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } \frac{u^j}{j^3} S_1
 =  
 4 H_{-1,0,0,1}(-y) 
+ \Snp{2,2}{y^2}
- 4 \Snp{2,2}{y} 
- 4 \Snp{2,2}{-y} 
- 6 \Li{4}{-y} 
\nonumber \\ && 
- 2 \Li{4}{y}
+ 4 \Snp{1,2}{-y}  \ln y 
+ 4 \Snp{1,2}{ y}  \ln y 
- 2 \Snp{1,2}{ y^2}  \ln(y)
+ 4 \Li{3}{-y} \ln (1-y) 
\nonumber \\ && 
+ 2 \Li{3}{-y} \ln y
+ 2 \Li{3}{y}  \ln y
- \Li{2}{y} \ln^2 y 
- 4 \Li{2}{-y} \ln y \ln (1-y) 
\nonumber \\ && 
- \tfrac{1}{3} \ln^3 y \ln (1-y) 
+ \tfrac{1}{24} \ln^4 y 
+ 2 \zeta_2 \Li{2}{y}
- \tfrac{1}{2} \zeta_2 \ln^2 y 
+ 2 \zeta_2 \ln y \ln (1-y) 
\nonumber \\ && 
+ 6 \zeta_3 \ln (1-y) 
- 3 \zeta_3 \ln y 
- \tfrac{7}{3} \zeta_3 \ln 2 
- 4 \zeta_4 \; , 
\\ && \hspace*{-10mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{AN_Sb1_3}
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } \frac{u^j}{j^3} \bar{S}_1
 =  
 4 H_{-1,0,0,1}(-y) 
+ \Snp{2,2}{y^2}
- 8 \Snp{2,2}{y} 
- 4 \Snp{2,2}{-y} 
- 6 \Li{4}{-y} 
\nonumber \\ && 
+ 2 \Li{4}{y}
- \left[ \Li{2}{y} \right]^2
+ 4 \Snp{1,2}{-y}  \ln y 
+ 8 \Snp{1,2}{ y}  \ln y 
- 2 \Snp{1,2}{ y^2} \ln y 
+ \tfrac{1}{48} \ln^4 y 
\nonumber \\ && 
+ 4 \Li{3}{-y} \ln (1-y) 
- 4 \Li{3}{y} \ln (1-y) 
+ 2 \Li{3}{-y} \ln y 
- 4 \Li{2}{-y} \ln y \ln (1-y) 
\nonumber \\ && 
+ 2 \Li{2}{y} \ln y \ln (1-y) 
- \tfrac{1}{2} \Li{2}{y} \ln^2 y 
- \tfrac{1}{6} \ln^3 y \ln (1-y) 
+ 4 \zeta_2 \ln y \ln (1-y) 
\nonumber \\ && 
- \zeta_2 \ln^2 y 
+ 10 \zeta_3 \ln (1-y) 
- 5 \zeta_3 \ln y 
+ 4 \zeta_2 \Li{2}{y}
- \tfrac{7}{3} \zeta_3 \ln 2 
- \tfrac{19}{2} \zeta_4 \; , 
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the factor $\sigma$ appears due to
$
%\frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}} = 
\tan\tfrac{\theta}{2} =
{\rm i} \sigma \frac{1-y}{1+y} \;$
and the combinations of harmonic sums
$C_j$  ($j=0,1,2$) are defined in~(\ref{combinations2}).
We have also used here the relations~(\ref{Hrel1}) and (\ref{Hrel2}).

The results for 
lower values of $c$ can be deduced using
\begin{equation}
\sum_{j=1}^\infty  \frac{u^j}{j^c} f(j) =
u \frac{{\rm d}}{{\rm d}u} \sum_{j=1}^\infty \frac{u^j}{j^{c+1}} f(j) \;. 
\label{derivative}
\end{equation}
For example, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray} 
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^2} S_1
&=& 
4 \Li{3}{-y} - 2 \Li{2}{-y} \ln y - \tfrac{1}{6} \ln^3 y
+ 3 \zeta_3  + \zeta_2 \ln y \; ,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\  
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j \right) } \frac{u^j}{j^2} \bar{S}_1
&=& 
-2 \Li{3}{y} + 4 \Li{3}{-y} - 2 \Li{2}{-y} \ln y + \Li{2}{y}\ln y
\nonumber \\ && 
- \tfrac{1}{12} \ln^3 y + 2 \zeta_2 \ln y + 5 \zeta_3 \;.
\end{eqnarray}
The results for these two sums can be extracted from Ref.~\cite{FKV99}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Application to Feynman diagrams}
\setcounter{equation}{0}


Below  we present results for the $\ep$-expansion of one- and two-loop 
master integrals shown in Fig.~1. 
In the rest of this paper we use the notation $u=p^2/m^2$.

\begin{figure}[th]
\begin{center}
\centerline{\vbox{\epsfysize=30mm \epsfbox{inverse_diag2.eps}}}
\caption{\label{propagator} One- and two-loop diagrams
considered in the paper.
Bold and thin lines correspond to massive and
massless propagators, respectively.}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{One-loop vertex}

Let us consider a one-loop triangle diagram with
$m_1=m_2=m_3\equiv m$, $p_1^2=p_2^2=0$, with an arbitrary (off-shell)
value of $p_3^2\equiv p^2$. Such diagrams occur, for example, in Higgs decay
into two photons or two gluons via a massive quark loop.
Following the notation of Ref.~\cite{BD-TMF}, we will denote
this integral (with unit powers of propagators) as
$J_3(1,1,1; m)$. According to Eq.~(40) of Ref.~\cite{BD-TMF}, the result
in an arbitrary space-time dimension $n=4-2\ep$ is
\begin{equation}
J_3(1,1,1; m) \left. \right|_{p_1^2 = p_2^2 = 0} =
- \tfrac{1}{2} \mbox{i} \pi^{2-\ep} (m^2)^{-1-\ep} \Gamma(1+\ep)\;
{}_{3}F_2 \left(\begin{array}{c|} 1, 1, 1+\ep \\
\frac{3}{2}, 2 \end{array} ~\frac{p^2}{4m^2} \right) \; .
\end{equation}
For this integral a number of one-fold integral representation are 
available, see Eqs.~(3.10)--(3.11) in Ref.~\cite{DK01}.
Expanding in $\ep$ we get
\begin{eqnarray}
\label{J3sums}
&& \hspace{-5mm}
J_3(1,1,1; m) \left. \right|_{p_1^2 = p_2^2 = 0} 
= - \mbox{i} \pi^{2-\ep} \Gamma(1+\ep)\; \frac{(m^2)^{-\ep}}{p^2}
\sum_{j=1}^{\infty} \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^2} 
\left[ 1 + \ep S_1
+ \tfrac{1}{2}\ep^2 \left( S_1^2 -S_2 \right)
\right.
\nonumber \\  && \hspace{8mm}
+ \tfrac{1}{6}\ep^3 \left( S_1^3 -3 S_1 S_2 + 2 S_3 \right)
+ \tfrac{1}{24}\ep^4 \left( S_1^4 -6 S_1^2 S_2 + 8 S_1 S_3 + 3 S_2^2 - 6 S_4 \right)
\left. 
+ {\cal O} (\ep^5) 
\right] .
\hspace*{9mm}
\end{eqnarray}
Substituting results for the occurring inverse binomial sums, we
obtain
\begin{eqnarray}
J_3(1,1,1; m) \left. \right|_{p_1^2 = p_2^2 = 0} 
&\!\!=\!\!& - \mbox{i} \pi^{2-\ep} \Gamma(1+\ep)\; \frac{(m^2)^{-\ep}}{p^2}
\Bigl\{ 
 - \tfrac{1}{2} \ln^2 y 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ &&
+ \ep \left[ 
4 \Li{3}{-y}
- 2 \Li{2}{-y} \ln y 
- \tfrac{1}{6} \ln^3 y 
+ \zeta_2 \ln y 
+ 3 \zeta_3 
\right]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && 
+ \ep^2 \Bigl[ 
2 \left[\Li{2}{-y} \right]^2
+ 2 \Li{3}{-y} \ln y 
- 4 \Snp{1,2}{-y} \ln y 
- \Li{2}{-y} \ln^2 y 
\nonumber \\ && 
+ 2 \zeta_2 \Li{2}{-y}
+ \tfrac{1}{2}\zeta_2 \ln^2 y 
+ 2 \zeta_3 \ln y 
- \tfrac{1}{24} \ln^4 y 
+ \tfrac{5}{4} \zeta_4
\Bigl]
+ {\cal O}(\ep^3) 
\Bigl\} \; . 
\label{Hgg1}
\end{eqnarray}
These results correspond to the analytic continuation of
Eqs.~(3.13), (3.14) and (3.16) from Ref.~\cite{DK01}.

The $\ep^3$ term is Eq.~(\ref{J3sums}) contains the same
combination of sums as $C_0$ in Eq.~(\ref{combinations2}).
However, the sum in Eq.~(\ref{J3sums}) contains $j^2$
in the denominator, i.e., it corresponds to the case $c=2$.
Using the result~(\ref{combination:1}) (with $c=1$) 
together with (\ref{higher_c}), we obtain a one-fold
integral representation for the $\ep^3$-term in Eq.~(\ref{J3sums}). 
Analyzing it, we see that we get only one new non-trivial integral,  
\begin{equation}
\label{nontrivial5}
\int\limits_0^\theta {\mbox d} \phi \; 
\Ls{2}{\phi} \; \ln^2 \left| 2\sin\tfrac{\phi}{2} \right| \; ,
\end{equation}
while all other terms can be expressed in terms of known functions.
For $\theta=\tfrac{2\pi}{3}$ the integral~(\ref{nontrivial5}) 
is connected with the new element $\chi_5$ of the {\em odd} basis,
whereas for $\theta=\tfrac{\pi}{2}$ it is related to
the new element $\widetilde{\chi}_5$ 
of the {\em even} basis, see Section~3.3 in Ref.~\cite{DK01} 
and Eqs.~(14)--(15) in Ref.~\cite{review}.

At the same time, the $\ep^4$-term of Eq.~(\ref{J3sums})
(and all other even powers of $\ep$)
can be calculated in terms of log-sine functions,
so that their analytic continuation can be expressed in terms of 
Nielsen polylogarithms.
Namely, for the $\ep^4$-term, combining 
Eqs.~(3.14), (3.19) from Ref.~\cite{DK01} and Eq.~(\ref{S4G}),
we obtain the result for the sum 
\[
\sum_{j=1}^{\infty} \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^c} 
\left( S_1^4 -6 S_1^2 S_2 + 8 S_1 S_3 - 3 S_2^2\right)
\]
with $c=2$. Using Eq.~(\ref{derivative}), we also obtain
the result for the case $c=1$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Two-loop self-energy integral $F_{10101}$}

This integral is a good illustration of the
application of general expressions given in Section~3.
The off-shell result for this integral in arbitrary
dimension was presented in~\cite{BFT93} (where it was called
$\widetilde{I}_3$, see Eq.~(22) of \cite{BFT93}).
For unit powers of propagators, the result reads
\begin{eqnarray}
&& \hspace*{-15mm}
m^{2+4\ep}
(1-2\ep) F_{10101}(p^2,m) =
\frac{1}{(1-\ep^2)(1+2\ep)}
{}_{4}F_3 \left(\begin{array}{c|} 1, 1+\ep,1+\ep, 1+2\ep \\
\frac{3}{2} +\ep ,2+\ep, 2-\ep \end{array} ~\frac{p^2}{4m^2} \right)
\nonumber\\ && \hspace*{-15mm}
-\frac{1}{2 \ep (1+\ep)}
{}_{3}F_2 \left(\begin{array}{c|} 1, 1\!+\!\ep, 1\!+\!\ep \\
\frac{3}{2}, 2+\ep \end{array} ~\frac{p^2}{4m^2} \right)
+ \frac{1}{2\ep}
\frac{\Gamma^2(1-\ep)}{\Gamma(1\!-\!2\ep)} \left(-\frac{m^2}{p^2}\right)^\ep
{}_{3}F_2 \left(\begin{array}{c|} 1, 1, 1+\ep \\
\frac{3}{2}, 2 \end{array} ~\frac{p^2}{4m^2} \right). 
\end{eqnarray}
Expanding in $\ep$, we obtain
\begin{eqnarray}
&& \hspace*{-10mm}
m^{2+4\ep}
(1-2\ep) F_{10101}(p^2,m) 
\nonumber\\ && \hspace*{-10mm}
=
\sum_{j=1}^\infty \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^2} 
\Biggl\{ 
\frac{3}{j} - \ln(-u)
+ \ep \Biggl[
- \frac{1}{j^2}
+ \frac{11}{j} S_1 
- \frac{4}{j} \bar{S}_1 
- \ln (-u) S_1 
+ \tfrac{1}{2} \ln^2 (-u) 
- \zeta_2 
\Biggr]
+ {\cal O} (\ep^2)\Biggr \}
\nonumber\\ && \hspace*{-10mm}
=  6 \Li{3}{y} - 6 \Li{2}{y} \ln y  -2 \ln^2 y \ln (1-y) - 6 \zeta_3 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber\\ && \hspace*{-10mm}
+ \ep \Bigl\{ 
  28  H_{-1,0,0,1}(-y) 
+ 7 \Snp{2,2}{y^2}
- 28 \Snp{2,2}{-y}
- 16 \Snp{2,2}{y}
- 42 \Li{4}{-y} 
- 26 \Li{4}{y}
\nonumber\\ && \hspace*{-10mm}
+ 4 \left[\Li{2}{y} \right]^2
+ 16 \Snp{1,2}{y} \ln y 
- 14 \Snp{1,2}{y^2} \ln y 
+ 28 \Snp{1,2}{-y} \ln y 
+ 18 \Li{3}{-y} \ln y 
\nonumber\\ && \hspace*{-10mm}
+ 20 \Li{3}{y} \ln y 
+ 20 \Li{3}{-y} \ln (1-y) 
+ 12 \Li{3}{y} \ln (1-y) 
- 2 \Li{2}{-y} \ln^2 y 
- 9 \Li{2}{y} \ln^2 y 
\nonumber\\ && \hspace*{-10mm}
- 24 \Li{2}{-y} \ln y \ln (1-y) 
- 4 \Li{2}{y} \ln y \ln (1-y)
- 2 \ln^3 y \ln (1-y) 
+ 6 \zeta_2 \Li{2}{y}
\nonumber\\ && \hspace*{-10mm} 
+ 4 \zeta_2 \ln y \ln (1-y) 
- 12 \zeta_3 \ln y 
+ 24 \zeta_3 \ln (1-y) 
- \tfrac{49}{3} \zeta_3 \ln 2 
- 9 \zeta_4 
\Bigr\}
+ {\cal O} (\ep^2) \; .
\label{f10101}
\end{eqnarray}
The result for the finite part coincides with~\cite{B90},
whereas the result for the $\ep$-term is new.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%J011
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Two-loop sunset-type diagram $J_{011}$}

Let us consider sunset-type diagrams with two equal masses and 
one zero mass (see Fig.~1). The off-shell result for the sunset-type 
integral $J_{011}$ with arbitrary powers of propagators 
has been obtained in Refs.~\cite{BFT93,D91}, by using the 
Mellin--Barnes technique \cite{BD-TMF}:
\begin{eqnarray}
J_{011}(\sigma,\nu_1,\nu_2;p^2,m) \!\! &=& \!\!
(m^2)^{n-\sigma-\nu_1-\nu_2}
\frac{
\Gamma(\nu_1\!+\!\nu_2\!+\!\sigma\!-\!n)
\Gamma\left(\frac{n}{2}\!-\!\sigma\right)
\Gamma\left(\nu_2\!+\!\sigma\!-\!\frac{n}{2}\right)
\Gamma\left(\nu_1\!+\!\sigma\!-\!\frac{n}{2}\right)}
{\Gamma(\nu_1)\Gamma(\nu_2)\Gamma\left(\frac{n}{2}\right)
\Gamma(\nu_1+\nu_2+2\sigma-n) \Gamma^2 \left(3-\frac{n}{2}\right)}
\nonumber\\ && \times
{}_{4}F_3 \left( \begin{array}{c|}
\sigma,\nu_1+\nu_2+\sigma-n,
\nu_2+\sigma-\tfrac{n}{2}, \nu_1+\sigma-\tfrac{n}{2} \\
\tfrac{n}{2}, \sigma + \tfrac{1}{2}(\nu_1+\nu_2-n),
\sigma+ \tfrac{1}{2}(\nu_1+\nu_2+1-n)
\end{array}~\frac{p^2}{4 m^2} \right) .
\label{j011_off-shell}
\end{eqnarray}
%
For simplicity, 
in the definition of the integral $J_{011}$
we will omit the arguments $p^2$ and $m$, 
$$
J_{011}(\sigma,\nu_1,\nu_2)  \equiv J_{011}(\sigma,\nu_1,\nu_2;p^2,m).
$$
Let us remind that for the integrals $J_{011}$ with 
different integer values of $\sigma$ and $\nu_i$
there are two master integrals~\cite{T97a} of this type,
$J_{011}(1,1,1)$ and $J_{011}(1,1,2)$.
However, two other independent combinations
of the integrals of this type happen to be more suitable for 
constructing the $\ep$-expansion, $J_{011}(1,2,2)$ and
$[J_{011}(1,2,2)+2J_{011}(2,1,2)$] (see also in Ref.~\cite{thresholds}). 
The latter combination corresponds to the integral $J_{011}(1,1,1)$ in 
$2-2\ep$ dimensions \cite{T96}.
In Ref.~\cite{single} 
$J_{011}(1,1,3)$ has been used as the second integral.
To construct the $\ep$-expansion of the integrals 
$J_{011}(1,1,1)$ and $J_{011}(1,1,2)$ up to 
order $\ep^2$, the integral $J_{011}(1,2,2)$ should also 
be expanded up to $\ep^2$, whereas
$[J_{011}(1,2,2)+2J_{011}(2,1,2)]$ or 
$J_{011}(1,1,3)$ up to the order $\ep$ only. 
In Appendix~C we give an example of a realistic calculation
which demonstrates the required orders of the $\ep$-expansion
for the integrals involved.

For $J_{011}(1,2,2)$ we have obtained
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
&& \hspace{-10mm}
J_{011}(1,2,2)  = 
\frac{(m^2)^{-1-2\ep}}{(1-\ep)(1+2\ep)}
{}_{3}F_2 \left( \begin{array}{c|}
1,  1+\ep, 1+2\ep \\
\tfrac{3}{2} + \ep, 2-\ep
\end{array}~\frac{p^2}{4 m^2} \right)  
\nonumber \\ && \hspace*{-5mm}
= 2 \frac{(m^2)^{-2\ep}}{p^2}
\sum_{j=1}^{\infty} \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^2} 
\Biggl\{ 
1 + \ep \Biggl[ \frac{1}{j} + 5 S_1 - 2 \bar{S}_1
\Biggr]
\nonumber \\ &&
+ \ep^2 \Biggl[ 
\frac{1}{j^2} 
+ \frac{5}{j}S_1 
- \frac{2}{j} \bar{S}_1
+ \tfrac{25}{2} S_1^2 
- 10 S_1 \bar{S}_1 
- \tfrac{5}{2} S_2
+ 2 \bar{S}_1^2 
+ 2 \bar{S}_2 
\Biggr]
+ {\cal O} (\ep^3) 
\Biggr\}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && \hspace*{-5mm}
= 2 \frac{(m^2)^{-2\ep}}{p^2} 
\left( 1-y \right)^{ 2 \ep}
y^{ 2\ep}
\Biggl\{ 
- \tfrac{1}{2} \ln^2 y 
\nonumber \\ &&
+ \ep 
\Bigl[ 
 \tfrac{1}{2} \ln^3 y 
+ \zeta_2 \ln y 
-6 \ln y  \Li{2}{-y} 
-4 \ln y  \Li{2}{y} 
+ 3 \zeta_3 
+ 12 \Li{3}{-y} 
+ 6 \Li{3}{y}
\Bigr]
\nonumber \\ &&
+ \ep^2 
\Bigl[  
 12 H_{-1,0,0,1} (-y)
-12 \ln y  \Snp{1,2}{-y} 
-12 \ln y \Snp{1,2}{y^2} 
+ 8 \ln y  \Snp{1,2}{y} 
- 12 \Snp{2,2}{-y}
\nonumber \\ &&
+ 3 \Snp{2,2}{y^2}
- 12 \ln (1-y) \Li{3}{-y} 
-\tfrac{7}{24} \ln^4 y 
- \tfrac{1}{2} \zeta_2 \ln^2 y 
+ 2 \zeta_3 \ln y 
+ 6 \ln y  \Li{3}{-y} 
\nonumber \\ && 
+ 8 \ln y  \Li{3}{y} 
- 7 \zeta_3 \ln 2 
+ 6 \zeta_2 \Li{2}{-y} 
+ 4 \zeta_2 \Li{2}{y}
+ \tfrac{13}{4} \zeta_4 
+ 18 \left[ \Li{2}{-y} \right]^2
\nonumber \\ && 
+ 12 \Li{2}{y} \Li{2}{-y}
+ 4 \left[ \Li{2}{y} \right]^2
- \tfrac{9}{4} \Li{4}{y^2}
\Bigr]
+ {\cal O} (\ep^3) 
\Biggr\} \; ,
\label{j011_122}
\end{eqnarray}
where the result for the $\ep^2$-term is new.

For another combination, we get
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
&& \hspace{-10mm}
J_{011}(1,2,2) + 2 J_{011}(2,1,2)
= - \frac{(m^2)^{-1-2\ep}}{\ep(1+2\ep)}
{}_{3}F_2 \left( \begin{array}{c|}
1,  1+\ep, 1+2\ep \\
\tfrac{3}{2} + \ep, 1-\ep
\end{array}~\frac{p^2}{4 m^2} \right)  
\nonumber \\ && \hspace*{-10mm}
= - \frac{2(m^2)^{-2\ep}}{p^2 } 
\sum_{j=1}^{\infty} 
\frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j}
\Biggl[ \frac{1}{\ep} 
+ \left( 5 S_1 - 2 \bar{S}_1 \right)
+ \ep \left(
\tfrac{25}{2} S_1^2 
- 10 S_1 \bar{S}_1 
- \tfrac{5}{2} S_2
+ 2 \bar{S}_1^2
+ 2 \bar{S}_2
\right)
\!+\! {\cal O} (\ep^2) 
\Biggr]
\nonumber \\ && \hspace*{-10mm}
= - \frac{2(m^2)^{-2\ep}}{p^2 }  
\left( 1+y \right)^{-1-6 \ep}
\left( 1-y \right)^{1-2 \ep}
\Biggl\{ 
\frac{1}{\ep}\ln y 
+ \Bigl[ 
2 \ln^2 y 
- \zeta_2 
- 6 \Li{2}{-y} 
- 2 \Li{2}{y}
\Bigr] 
\nonumber \\ && 
+ \ep \Bigl[ 
24 \Snp{1,2}{-y} 
\!+\! 6 \Snp{1,2}{y^2}
\!-\! 8 \Snp{1,2}{y} 
\!+\! \tfrac{5}{3} \ln^3 y 
\!-\! 4 \zeta_2 \ln y 
\!-\! 11 \zeta_3 
\!-\! 24 \Li{3}{-y} 
\!-\! 8 \Li{3}{y}
\Bigr] 
\nonumber \\ && 
+ {\cal O} (\ep^2) 
\Biggr\} \;.
\label{j011_c}
\end{eqnarray}
The $\ep$-term can be related to that of the result for
$J_{011}(1,1,3)$ presented in Ref.~\cite{single}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Two-loop vertex diagram $P_{126}$ } 

Consider the two-loop vertex-type diagram $P_{126}$ given 
in Ref.~\cite{FKV99}\footnote{Their $q^2$ corresponds to our $p^2$,
whereas their $z$ corresponds to our $u$.} (see Fig.~1),
\begin{eqnarray} 
(p^2)^2 P_{126}  &=& 
    \sum_{j=1}^{\infty} \frac{1}{\left( 2j \atop j\right) } \frac{u^j}{j^2}
      \Biggl\{
      \frac{1}{\varepsilon^2}
    + \frac{1}{\varepsilon} \Bigl[
           - S_1
           - \log(-u)
             \Bigr]
\nonumber \\ &&
      + \tfrac{1}{2} \log^2(-u)
      - S_1 \log(-u)
      - \tfrac{3}{2} S_2
      - \tfrac{15}{2} S_1^2
      + 4 S_1 \bar{S}_1  
      + 2 \frac{S_1}{j}
+ {\cal O} (\ep)
      \Biggr\} \; .
\end{eqnarray}
It corresponds to one of the two-loop contributions to 
a boson decay into two massless particles, 
with a massive triangle subloop. Diagrams of such type  
have been intensively studied for Higgs boson production 
via gluon fusion~\cite{Hgg}.

Using our approach, we obtain
\begin{eqnarray} 
(p^2)^2 P_{126}  &\!\!=\!\!& 
     - \frac{1}{2\ep^2} \ln^2 y 
+ \frac{1}{\ep} \Bigl[ 
2 \Li{2}{-y} \ln y 
\!-\! 4 \Li{3}{-y}
\!+\! \ln^2 y \ln (1-y) 
\!-\! \tfrac{1}{3} \ln^3 y 
\!-\! \zeta_2 \ln y 
\!-\! 3 \zeta_3 
\Bigr]
\nonumber \\ &&
+  8 H_{-1,0,0,1}(-y) 
+ 2 \Snp{2,2}{y^2}
- 8  \Snp{2,2}{-y}
- 8  \Snp{2,2}{y}
- 4 \Li{4}{y}
- 12 \Li{4}{-y} 
\nonumber \\ &&
- 8 \Li{2}{y} \Li{2}{-y} 
- 14 \left[ \Li{2}{-y} \right]^2
+ 28 \Snp{1,2}{-y} \ln y 
- 10 \Li{3}{-y} \ln y 
\nonumber \\ &&
+ 7 \Li{2}{-y} \ln^2 y 
- 4 \Li{2}{-y} \ln y \ln (1-y) 
- \ln^2 y \ln^2 (1-y) 
- \tfrac{1}{6} \ln^4 y 
\nonumber \\ &&
+ \tfrac{2}{3} \ln^3 y \ln (1-y) 
+ 2 \zeta_2 \ln y \ln (1-y) 
- \tfrac{5}{2} \zeta_2 \ln^2 y 
+ 6 \zeta_3 \ln (1-y) 
- 11 \zeta_3 \ln y
\nonumber \\ &&
- 6 \Li{2}{-y} \zeta_2 
- \tfrac{14}{3} \zeta_3 \ln 2 
- \tfrac{27}{4} \zeta_4 
+ {\cal O} (\ep) \; .
\label{Hgg2}
\end{eqnarray}
Here, the result for the finite part is new.
Alternatively, results of such type
%(involving the same harmonic polylogarithm)
can be obtained in a different way, using 
a technique based on Mellin--Barnes contour integrals~\cite{S_Bha}. 

\section{Conclusion}

In this paper, we have studied the
{\it multiple inverse binomial sums} of the type~(\ref{binsum}), 
for arbitrary values of the variable~$u$.
Our analysis was based on the connection between 
these sums and terms of the expansion of certain
hypergeometric functions with respect to the parameter~$\ep$.
Using known results for the hypergeometric functions,
together with the integral representation~(\ref{higher_c}),
we have obtained a number of new analytical results
up to the weight {\bf 4}, Eqs.~(\ref{S1a})--(\ref{last4}),
(\ref{S1c2})--(\ref{S2bar_2}) and (\ref{S1_3})--(\ref{S1bar_3}).
Moreover, in some cases like (\ref{S2G})--(\ref{S4G})
the results can be obtained for an arbitrary weight.
Constructing analytical continuation of the obtained results, 
we have expressed them in terms of the generalized
polylogarithms (\ref{AN_S1S1_2})--(\ref{AN_Sb1_3}).
In the cases considered, only one new function was needed,
in addition to the basis of Nielsen polylogarithms.
As such function one can take, e.g., the harmonic polylogarithm 
$H_{-1,0,0,1}(-y)$, where the variable $y$ is given in
Eq.~(\ref{y<->u}).

This approach allowed us to construct some terms of the 
$\ep$-expansion of the generalized hypergeometric function 
${}_{P+1}F_P$ (\ref{expansion}) and 
obtain new analytical results for higher terms of the 
$\ep$-expansion of some one- and two-loop 
propagator-type (see Eqs.~(\ref{f10101}), (\ref{j011_122}) and (\ref{j011_c})) 
and vertex-type (see Eqs.~(\ref{Hgg1}) and (\ref{Hgg2})) 
diagrams depending on one dimensionless variable $u=p^2/m^2$.
As a by-product, we have analytically proven the 
earlier published results for the three-loop vacuum integrals, 
the finite part of ${\bf D_4}$ and 
the $\ep$-part of ${\bf D_3}$,
corresponding to the particular value of $u=1$ ($z=\tfrac{1}{4}$),
see Sections~4.3 and 4.6 in Ref.~\cite{DK01} and Refs.~\cite{B99,odd,d_3}.
These integrals enter as master integrals in Avdeev's 
package~\cite{leo96} and {\sf MATAD}~\cite{matad}. 
Moreover, the developed technique is useful in 
the calculation of two-loop vertex-type diagrams~\cite{vertex}.

\vspace{5mm}


{\bf Acknowledgements.}
We are grateful to F.~Jegerlehner, M.~Spira, V.A.~Smirnov and
O.V.~Tarasov for useful discussions.
We would like to thank R.~Delbourgo and J.~Fleischer for their
interest in our work, and
J.~Gluza for checks of some formulae.
This research was supported in part by INTAS-CERN grant No.~99-0377
and by the Australian Research Council grant No.~A00000780.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Harmonic polylogarithms of complex arguments}
\label{HPCA}
\setcounter{equation}{0}

Here we collect some properties 
of the harmonic polylogarithms introduced in~\cite{RV00}. 
Let us define a $w$-dimensional vector $\vec{a}=(a,\vec{b})$, where
$a$ is the leftmost component of $\vec{a}$, while $\vec{b}$ stands
for the vector of the remaining $(w-1)$ components. The harmonic
polylogarithms of weight $w$ are then defined as follows:
\begin{equation}
\H_{\vec{a}}(y) = \int\limits_0^y \d x \ \f(a;x) \ \H_{\vec{b}}(x) \ ,
\label{eq:def}
\end{equation}
where the three rational fractions $\f(a;x)$ are given by
\[
   \f(+1;x) = \frac{1}{1-x} \;, \quad  
   \f(0;x)  = \frac{1}{x} \;, \quad  
   \f(-1;x) = \frac{1}{1+x} \; .
\]
The derivatives can be written in a compact form
\begin{equation}
\frac{\d}{\d y} \H_{\vec{a}}(y) = \f(a;y) \H_{\vec{b}}(y) \; .
\label{eq:derive}
\end{equation}

Let us put $y=e^{{\rm i}\theta}$, so that 
\begin{equation}
\H_{\vec{a}}(e^{{\rm i}\theta})  
= \H_{\vec{a}}(1) 
+ {\rm i} \int\limits_0^\theta \d \phi \ e^{{\rm i}\phi}\;
\f(a;e^{{\rm i}\phi}) 
\ \H_{\vec{b}}(e^{{\rm i}\phi})  \; ,
\end{equation}
where 
\begin{eqnarray}
   e^{{\rm i}\phi} \f(+1;e^{{\rm i}\phi}) &=& 
   \frac{e^{{\rm i}(\pi+\phi)/2}}{2 \sin \tfrac{\phi}{2} } 
   = - \frac{1}{2} 
\left( 1 - {\rm i} \cot \tfrac{\phi}{2}  \right)  
\; ,
\nonumber\\
   e^{{\rm i}\phi} \f(0;e^{{\rm i}\phi})  &=&  1 \; , 
\nonumber\\
   e^{{\rm i}\phi} \f(-1;e^{{\rm i}\phi}) &=& 
   \frac{e^{{\rm i}\phi/2}}{2 \cos \tfrac{\phi}{2} } 
   = \frac{1}{2} 
\left( 1 + {\rm i} \tan \tfrac{\phi}{2} \right) 
\; .
\end{eqnarray}

Consider, for example, the harmonic polylogarithm $H_{-1,0,0,1}(y)$
given in Eq.~(\ref{h(-1001)}).
Using the decomposition of 
$\Li{3}{e^{{\rm i}\phi}}$ into 
the real and imaginary parts 
\cite{Lewin} and integrating by parts, we obtain 
expressions in terms of Clausen's and generalized log-sine functions,
\begin{eqnarray}
\H_{-1,0,0,1}(e^{{\rm i}\theta})&\!\!=\!\!& 
\H_{-1,0,0,1}(1)  
+ \frac{{\rm i}}{2} \int\limits_0^\theta {\rm d} \phi \; \Li{3}{e^{{\rm i}\phi}}
\left( 1 + {\rm i} \frac{\sin \tfrac{\phi}{2}}{\cos \tfrac{\phi}{2} }  \right) 
\nonumber \\ 
&\!\!=\!\!& \H_{-1,0,0,1}(1)   
- \tfrac{1}{96} \theta^2 (2\pi-\theta)^2
+ \Phi(\theta) 
+ \ln \left(2 \cos \tfrac{\theta}{2} \right) \Cl{3}{\theta} - \zeta_3 \ln2 
\nonumber \\ &&
+ {\rm i} \ln \left(2 \cos \tfrac{\theta}{2} \right) \Gl{3}{\theta}
+ \tfrac{1}{2} {\rm i} \left[ \Cl{4}{\theta} + \Cl{4}{\pi-\theta} \right]
\nonumber \\ && 
- {\rm i} \left[ 
\Gl{2}{\theta}
\Cl{2}{\pi-\theta}
+ \tfrac{1}{2} (\pi- \theta) \Cl{3}{\pi-\theta}
- \tfrac{1}{2}\pi \Cl{3}{\pi}
\right] \;, 
\end{eqnarray}
where 
$$
\Cl{3}{\pi} = -\tfrac{3}{4} \zeta_3 \;, \quad
\Gl{2}{\theta} =  \zeta_2 - \tfrac{1}{2} \pi \theta + \tfrac{1}{4}\theta^2 \;,
\quad 
\Gl{3}{\theta} = \tfrac{1}{12}\theta \left(\pi-\theta \right) 
\left(2\pi-\theta \right) \; ,
$$
and $\Phi(\theta)$ is defined in Eq.~(\ref{def_Phi}).

In this paper, we have also used the following relations:
\begin{eqnarray}
\label{Hrel1}
H_{-1,0,0,1}\left( -y^{-1} \right) &=& H_{-1,0,0,1}(-y)
+ \Li{4}{-y^{-1}}
+ \Li{4}{y} 
- \Li{3}{y} \ln y 
- \zeta_2 \Li{2}{1-y} 
\nonumber \\ && 
+ \tfrac{1}{2} \ln^2 y \Li{2}{y} 
+ \tfrac{1}{6} \ln^3 y \ln (1-y)
- \tfrac{1}{8} \zeta_4
- {\rm i} \sigma \pi \tfrac{3}{4} \zeta_3 \; 
\end{eqnarray}
and 
\begin{equation}
\label{Hrel2}
H_{-1,0,0,1}(y) +  H_{-1,0,0,1}(-y) 
= \Li{2}{y} \Li{2}{-y}
+ \ln (1+y) \Li{3}{y}
+ \ln (1-y) \Li{3}{-y} \; .
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%         int_li3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The following two representations of $H_{-1,0,0,1}(y)$ are also useful:
\begin{eqnarray}
H_{-1,0,0,1}(y)
& = & \Li{4}{y}-{\textstyle{3\over4}} \zeta_3 \ln(1+y)
+ {\textstyle{1\over2}}
\int\limits_0^1 \frac{\mbox{d} x\; \ln^2{x}\; \ln(1-xy)}{1+x} \; , 
\\
H_{-1,0,0,1}(y)
& = & 
-{\textstyle{3\over4}} \zeta_3 \ln(1+y)
+ {\textstyle{1\over16}} \sum\limits_{l=1}^{\infty}
\frac{y^l}{l} \left[ \psi''\left(\frac{l+1}{2}\right)
-\psi''\left(\frac{l}{2}\right) \right] .
\end{eqnarray}

Instead of $\Phi(\theta)$, 
one could also introduce another function. One of such possibilities
is to consider a generalization of the Glashier function.
Let us recall that the real part of 
$\Li{j}{e^{{\rm i}\theta}}$ can be presented as\footnote{We use
the standard notation $\Li{j}{r,\theta}=
{\rm Re}\left[\Li{j}{re^{{\rm i}\theta}}\right]$
(see in Ref.~\cite{Lewin}).}
\begin{equation}
\Li{j}{1,\theta} =
\frac{(-1)^{j-1}}{2(j-2)!}\int\limits_0^1
\frac{{\rm d}\xi}{\xi}\; \ln^{j-2}\xi\; 
\ln\left(1-2\xi\cos\theta+\xi^2\right)
=
\Biggl\{
\begin{array}{l}
\Cl{j}{\theta}, \quad j\;\;{\rm odd}\\
\Gl{j}{\theta}, \quad j\;\;{\rm even}
\end{array}
\end{equation}
where $\Cl{j}{\theta}$ and $\Gl{j}{\theta}$ are
Clausen and Glashier functions, respectively.
In particular, $\Gl{j}{\theta}$ is just a polynomial
in $\theta$.

A possible non-trivial generalization of $\Gl{j}{\theta}$
(for even $j$) could be
\begin{equation}
\Gl{j}{\theta; a} =
-\frac{1}{2(j-2)!}\int\limits_0^1
\frac{{\rm d}\xi}{\xi+a}\; \ln^{j-2}\xi\; 
\ln\left(1-2\xi\cos\theta+\xi^2\right), \quad
j\;\;{\rm even},
\end{equation}
so that $\Gl{j}{\theta; 0}=\Gl{j}{\theta}$.
In particular,
\begin{equation}
\label{Gl2_gen}
\Gl{2}{\theta; a} = -\ln{b}\;\ln\frac{a+1}{a}
+ \Li{2}{\frac{a+1}{b},\widetilde{\theta}}
- \Li{2}{\frac{a}{b},\widetilde{\theta}}\; ,
\end{equation}
where
\begin{equation}
\cos{\widetilde{\theta}}
=\frac{a+\cos\theta}{\sqrt{1+2a\cos\theta+a^2}}, 
\end{equation}
so that
\begin{equation}
a = \frac{\sin(\theta-{\widetilde{\theta}})}
         {\sin{\widetilde{\theta}}}, \quad
b = \sqrt{1+2a\cos\theta+a^2} 
= \frac{\sin\theta}{\sin{\widetilde{\theta}}} \; .
\end{equation}
Using Eqs.~(17) and (18) on p.~293 of~\cite{Lewin},
we can see that the general result (\ref{Gl2_gen})
simplifies in the case $a=1$ (${\widetilde{\theta}}=\tfrac{\pi}{2}$),
\begin{equation}
\Gl{2}{\theta; 1} = 
-\tfrac{1}{4}\; \Li{2}{\cos^2{\tfrac{\theta}{2}}}
+ \tfrac{1}{8}(\pi-\theta)^2 
- \tfrac{1}{2} \ln^2{2} \; .
\end{equation}

Then, let us consider
\begin{equation}
\label{Gl4(theta,1)}
\Gl{4}{\theta; 1} = -\frac{1}{4}
\int\limits_0^1 \frac{{\rm d}\xi}{\xi+1}\;
\ln^2\xi\; \ln\left(1-2\xi\cos\theta+\xi^2\right) \; . 
\end{equation}
For $\theta=0$ we get
\begin{equation}
\Gl{4}{0; 1} = U_{3,1} - \zeta_4 \; ,
\end{equation}
where $U_{3,1}$ is the alternating two-fold Euler sum 
considered in~\cite{euler,B99} (see also Ref.~\cite{GKP}), 
$$
U_{3,1} = 
- 2 \Li{4}{\tfrac{1}{2}} + \tfrac{1}{2} \zeta_4
- \tfrac{1}{12} \ln^4 2 + \tfrac{1}{2} \zeta_2 \ln^2 2 \;. 
$$

For general $\theta$,
one can see that $\Gl{4}{\theta;1}$ is related to the real part
of a harmonic polylogarithm,
\begin{equation} 
{\rm Re}H_{-1,0,0,1}\big(e^{{\rm i}\theta}\big)
= \Gl{4}{\theta} - \Gl{4}{\theta; 1}
- \tfrac{3}{4} \zeta_3 \ln\left(2\cos\tfrac{\theta}{2}\right).
\end{equation}
In particular, the function $\Phi(\theta)$
can be presented as
\begin{equation}
\Phi(\theta) = -\Gl{4}{\theta; 1} + \Gl{4}{0; 1}
+\tfrac{7}{4}\zeta_3\ln{2} - \tfrac{1}{96}\theta^2(2\pi-\theta)^2
-\left[\Cl{3}{\theta}-\Cl{3}{\pi}\right]
\ln\left(2\cos\tfrac{\theta}{2}\right) ,
\end{equation}
where $\Cl{3}{\pi}=-\tfrac{3}{4}\zeta_3$.

Using the symmetry property of $\Phi(\theta)$, Eq.~(\ref{sym_Phi}), 
we get
\begin{eqnarray}
\Gl{4}{\theta; 1} + \Gl{4}{\pi-\theta; 1} &=&
\tfrac{11}{8}\zeta_4 - \tfrac{1}{48}\theta^2(\pi-\theta)^2
-\Cl{2}{\theta}\;\Cl{2}{\pi-\theta}
\nonumber \\ &&
-\left[\Cl{3}{\theta}-\Cl{3}{\pi}\right]
\ln\left(2\cos\tfrac{\theta}{2}\right)
\nonumber \\ &&
-\left[\Cl{3}{\pi-\theta}-\Cl{3}{\pi}\right]
\ln\left(2\sin\tfrac{\theta}{2}\right).
\label{sym_Gl4}
\end{eqnarray}
For special values of $\theta$, Eq.~(\ref{sym_Gl4}) yields
\begin{eqnarray}
\Gl{4}{\pi; 1} &=& -U_{3,1} + \tfrac{19}{8}\zeta_4
-\tfrac{7}{4}\zeta_3 \ln{2} ,
\\
\Gl{4}{\tfrac{\pi}{2}; 1} &=& \tfrac{161}{256}\zeta_4
-\tfrac{21}{64}\zeta_3\ln{2} - G^2 , 
\\
\Gl{4}{\tfrac{\pi}{3}; 1}+\Gl{4}{\tfrac{2\pi}{3}; 1}
&=& \tfrac{277}{216}\zeta_4 - \tfrac{13}{24}\zeta_3\ln{3}
-\tfrac{2}{3}\left[ \Cl{2}{\tfrac{\pi}{3}} \right]^2 \; ,
\end{eqnarray}
where $G$ is the Catalan constant.
Moreover, with the help of {\sf PSLQ} algorithm \cite{PSLQ} one can
obtain results for $\Gl{4}{\tfrac{\pi}{3}; 1}$ and 
$\Gl{4}{\tfrac{2\pi}{3}; 1}$ separately,
\begin{eqnarray}
\Gl{4}{\tfrac{\pi}{3}; 1}
&=& \tfrac{259}{108} \zeta_4
- \tfrac{13}{24} \zeta_3 \ln{3}
-\tfrac{1}{3}\left[ \Cl{2}{\tfrac{\pi}{3}} \right]^2
+ \tfrac{1}{4} \pi \Ls{3}{\tfrac{2\pi}{3}}
- \tfrac{3}{8} \LS{4}{1}{\tfrac{2\pi}{3}} \; ,
\\
\Gl{4}{\tfrac{2\pi}{3}; 1}
&=& -\tfrac{241}{216} \zeta_4
-\tfrac{1}{3}\left[ \Cl{2}{\tfrac{\pi}{3}} \right]^2
- \tfrac{1}{4} \pi \Ls{3}{\tfrac{2\pi}{3}}
+ \tfrac{3}{8} \LS{4}{1}{\tfrac{2\pi}{3}} \; .
\end{eqnarray}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Additional identities between inverse binomial sums}
\label{AIBIBN}
\setcounter{equation}{0}
%=====================================================================
As we have seen, the $\ep$-expansion of the
hypergeometric functions
produces series of the type (\ref{binsum}).
One can use certain properties of hypergeometric functions
to get relations between the sums (\ref{binsum}) involved
in the $\ep$-expansion. 

Let us consider Eq.~(B.18) of Ref.~\cite{DK01}
(which follows from Eq.~(22) on p.~498 of~\cite{PBM3}),
\begin{equation}
{}_{3}F_2 \left(\begin{array}{c|}
1+a_1 \ep, 1+ a_2 \ep, 1+ \tfrac{1}{2}(a_1+a_2)\ep  \\
\tfrac{3}{2}+ \tfrac{1}{2}(a_1+a_2)\ep, 2+(a_1+a_2)\ep \end{array} ~z
\right) =
(1\!-\!z) \left[
{}_{2}F_1 \left(\begin{array}{c|} 1+\tfrac{1}{2} a_1 \ep,
1+\tfrac{1}{2} a_2 \ep \\
\tfrac{3}{2}+\tfrac{1}{2}(a_1+a_2)\ep \end{array} ~z \right)
\right]^2 .
\label{less_trivial}
\end{equation}
It reduces the given $_3F_2$ function to a square of the $_2F_1$
function.
Substituting the $\ep$-expansions of the $_3F_2$ and $_2F_1$ functions 
into (\ref{less_trivial}), we
obtain the following relations between the sums:
\begin{eqnarray}
&&
\Sigma_{-;-;2}^{-;-}(u) = \frac{4-u}{2u} 
\left[ \Sigma_{-;-;1}^{-;-}(u)\right]^2 \; ,
\\ &&
\Sigma_{1;-;2}^{1;-}(u) - \Sigma_{-;1;2}^{-;1}(u) - \Sigma_{-;-;3}^{-;-}(u)
= \frac{4-u}{u} \; \Sigma_{-;-;1}^{-;-}(u)
\left[
\Sigma_{1;-;1}^{1;-}(u) - \Sigma_{-;1;1}^{-;1}(u) \right] \; ,
\\ &&
\Sigma_{2;-;2}^{1;-}(u) = \frac{4-u}{4u} \; \Sigma_{-;-;1}^{-;-}(u)
\Sigma_{2;-;1}^{1;-}(u) \; ,
\\ &&
\Sigma_{1;-;2}^{2;-}(u) - 2 \Sigma_{1;1;2}^{1;1}(u)
+\Sigma_{-;1;2}^{-;2}(u) + \Sigma_{-;2;2}^{-;1}(u) 
-2\Sigma_{1;-;3}^{1;-}(u) + 2\Sigma_{-;1;3}^{-;1}(u)
+2\Sigma_{-;-;4}^{-;-}(u) 
\nonumber \\ && \hspace*{10mm}
= \frac{4-u}{u}
\Bigl[
\Sigma_{-;-;1}^{-;-}(u)
\left( \Sigma_{1;-;1}^{2;-}(u) - 2 \Sigma_{1;1;1}^{1;1}(u)
+\Sigma_{-;1;1}^{-;2}(u) + \Sigma_{-;2;1}^{-;1}(u) 
-\tfrac{3}{8} \Sigma_{2;-;1}^{1;-}(u)\right)
\nonumber \\ && \hspace*{24mm}
+ \left( \Sigma_{1;-;1}^{1;-}(u) - \Sigma_{-;1;1}^{-;1}(u)\right)^2 
\Bigl] \; ,
\end{eqnarray}
with $u=4z$, so that
\[
\frac{4-u}{u} = \frac{1-z}{z} 
= \cot^2\tfrac{\theta}{2} \; . 
\]

Another interesting relation, Eq.~(B.19) of Ref.~\cite{DK01} 
(which follows from Eq.~(20) on p.~498 of~\cite{PBM3}), reads
\begin{eqnarray}
&& \hspace*{-10mm}
{}_{3}F_2 \left(\begin{array}{c|}
1+a_1 \ep, 1+ a_2 \ep, 1+ \tfrac{1}{2}(a_1+a_2)\ep  \\
\tfrac{3}{2}+ \tfrac{1}{2}(a_1+a_2)\ep, 1+(a_1+a_2)\ep \end{array} ~z
\right) =
{}_{2}F_1 \left(\begin{array}{c|} 1+\tfrac{1}{2}a_1\ep,
1+\tfrac{1}{2} a_2 \ep \\
\tfrac{3}{2}+\tfrac{1}{2}(a_1+a_2)\ep \end{array} ~z \right)
\nonumber \\ &&     
\times \left\{ 1 + \frac{a_1 a_2 \ep^2 z}
{2\left[1+(a_1+a_2)\ep\right]}\;
{}_{3}F_2 \left(\begin{array}{c|}
1+\tfrac{1}{2} a_1 \ep, 1+\tfrac{1}{2} a_2 \ep, 1 \\
\tfrac{3}{2}+ \tfrac{1}{2}(a_1+a_2)\ep, \; 2 \end{array} ~z \right)
\right\} \; .
\label{even_less_trivial}
\end{eqnarray}
In this way, we get the following relations:
\begin{eqnarray}
&&
\Sigma_{2;-;1}^{1;-}(u) = \tfrac{1}{3}
\Sigma_{-;-;1}^{-;-}(u) \Sigma_{-;-;2}^{-;-}(u) \; ,
\nonumber \\
&& 
3\Sigma_{1,2;-;1}^{1,1;-}(u)
-3\Sigma_{2,1;1}^{1;1}(u) 
- \tfrac{7}{2} \Sigma_{3;-;1}^{1;-}(u)
\nonumber \\ && \hspace*{20mm}
= \Sigma_{-;-;1}^{-;-}(u)
\left[ \Sigma_{1;-;2}^{1;-}(u) - \Sigma_{-;1;2}^{-;1}(u) \right]
+ \Sigma_{-;-;2}^{-;-}(u)
\left[ \Sigma_{1;-;1}^{1;-}(u) - \Sigma_{-;1;1}^{-;1}(u) \right] \; .
\hspace*{7mm}
\end{eqnarray}
We have checked that all above equations are satisfied by
the explicit results for the sums 
listed in this paper.

Let us also present some relations for higher-order sums:
\begin{eqnarray}
\label{higher1}
&& \hspace{-5mm}
\Sigma_{3;-;2}^{1;-}(u) 
+ \Sigma_{2;-;3}^{1;-}(u) 
- \Sigma_{2,1;-;2}^{1,1;-}(u) 
+ \Sigma_{2;1;2}^{1;1}(u) 
= 
\frac{4\!-\!u}{4u}\! 
\left\{
\Sigma_{2;-;1}^{1;-}(u)
\left[ 
  \Sigma_{-;1;1}^{-;1}(u) 
- \Sigma_{1;-;1}^{1;-}(u) 
\right]
\right.
\nonumber \\ && 
\left.
+ 
\Sigma_{-;-;1}^{-;-}(u) 
\left[ 
  \Sigma_{2;1;1}^{1;1}(u) 
- \Sigma_{2,1;-;1}^{1,1;-}(u) 
+ \tfrac{1}{2} \Sigma_{3;-;1}^{1;-}(u) 
\right] 
\right\} , 
\\ 
\label{higher2}
&& \hspace{-5mm}
  \Sigma_{2;-;2}^{2;-}(u) 
- \Sigma_{4;-;2}^{1;-}(u) 
=
\frac{4-u}{16u} 
\left\{  
\left[   \Sigma_{2;-;1}^{1;-}(u) \right]^2
+  \Sigma_{-;-;1}^{-;-}(u)  \left[
  \Sigma_{2;-;1}^{2;-}(u) 
- \Sigma_{4;-;1}^{1;-}(u)   \right]
\right\} \; ,
\nonumber \\ 
\label{higher3}
&& \hspace{-5mm}
\Sigma_{2;-;1}^{2;-}(u) 
-\Sigma_{4;-;1}^{1;-}(u) 
= \tfrac{2}{45} \Sigma_{-;-;1}^{-;-}(u) 
\left\{
\left[ \Sigma_{-;-;2}^{-;-}(u) \right]^2
+ 3 \Sigma_{2;-;2}^{1;-}(u) 
\right\} \;, 
\\
\label{higher4}
&& \hspace{-5mm}
2 \left[\Sigma_{-;-;2}^{-;-}(u)  \right]^2
\left[ 
  \Sigma_{1;-;1}^{1;-}(u) 
- \Sigma_{-;1;1}^{-;1}(u) 
\right]
+ 4  \Sigma_{-;-;2}^{-;-}(u)  \Sigma_{-;-;1}^{-;-}(u) 
\left[ 
  \Sigma_{1;-;2}^{1;-}(u) 
- \Sigma_{-;1;2}^{-;1}(u) 
\right] 
\nonumber \\ && \hspace{-5mm}
+ 6 \Sigma_{2;-;2}^{1;-}(u) 
\left[ 
  \Sigma_{1;-;1}^{1;-}(u) 
- \Sigma_{-;1;1}^{-;1}(u) 
\right]
+ 6  \Sigma_{-;-;1}^{-;-}(u) 
\left[ 
  \Sigma_{2,1;-;2}^{1,1;-}(u) 
- \Sigma_{2;1;-}^{1;1}(u) 
\right] 
\nonumber \\ && \hspace{-5mm}
+ 45  \Sigma_{2;1;1}^{2;1}(u) 
- 45 \Sigma_{2,1;-;1}^{2,1;-}(u) 
+ 45 \Sigma_{4,1;-;1}^{1,1;-}(u) 
- 45 \Sigma_{4;1;1}^{1;1}(u) 
+ 93 \Sigma_{3,2;-;1}^{1,1;-}(u) 
- 93 \Sigma_{5;-;1}^{1;-}(u) 
\nonumber \\ && \hspace{-5mm}
+ 4 \Sigma_{3;-;1}^{1;-}(u)  \Sigma_{-;-;2}^{-;-}(u) 
- 3 \Sigma_{3;-;2}^{1;-}(u)  \Sigma_{-;-;1}^{-;-}(u)  
= 0 \; , 
\end{eqnarray}
where Eqs.~(\ref{higher1}) and (\ref{higher2}) follow 
from Eq.~(\ref{less_trivial}),
while Eqs.~(\ref{higher3}) and (\ref{higher4}) follow 
from Eq.~(\ref{even_less_trivial}).

One more relation can be derived from Eq.~(47) 
on p.~456 of~\cite{PBM3},
\begin{equation}
{}_{2}F_1 \left(\begin{array}{c|}
1+a_1 \ep, 1+ a_2 \ep \\
\tfrac{3}{2}+ \tfrac{1}{2}(a_1+a_2)\ep 
\end{array} ~\sin^2\tfrac{\theta}{2}
\right) =
\cos\theta \;\;
{}_{2}F_1 \left(\begin{array}{c|} 1+\tfrac{1}{2} a_1 \ep,
1+\tfrac{1}{2} a_2 \ep \\
\tfrac{3}{2}+\tfrac{1}{2}(a_1+a_2)\ep \end{array} ~\sin^2\theta \right) .
\label{less_trivial2}
\end{equation}
We can compare the $\ep$-expansions of these $_2F_1$ functions,
considering the coefficients of $\ep^k$ as functions of $\theta$.
Introducing
\[
u=4\sin^2\tfrac{\theta}{2}, \qquad
{\widetilde{u}}=4\sin^2\theta, \qquad
A_1=a_1+a_2, \qquad A_2=a_1^2+a_2^2 \; ,
\]
we get the following relations at orders $\ep^0$ and $\ep^1$:
\begin{eqnarray}
\cot \tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j}
&=& \tfrac{1}{2} 
\cot \theta
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}
\frac{{\widetilde{u}}^j}{j} \; ,
\\
\cot \tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j}
\left( \tfrac{3}{2}S_1 - \bar{S}_1 \right)
&=&\tfrac{1}{2} 
\cot \theta
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}
\frac{{\widetilde{u}}^j}{j} 
\left( S_1 - \bar{S}_1 \right) \; .
\end{eqnarray}
At order $\ep^2$, comparison of the coefficients 
of $A_2$ and $A_1^2$ yields
\begin{eqnarray}
&&
\cot \tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j} S_2
=\tfrac{1}{8} 
\cot \theta
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}
\frac{{\widetilde{u}}^j}{j} S_2 \; ,
\\
&&
\cot \tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j}
\left[ \tfrac{1}{2}\left(\bar{S}_2+\bar{S}_1^2\right) 
-\tfrac{3}{2} S_1 \bar{S}_1 
-\tfrac{1}{8}S_2 +\tfrac{9}{8}S_1^2 \right]
\nonumber \\ && \hspace*{30mm}
=\tfrac{1}{2} 
\cot \theta
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}
\frac{{\widetilde{u}}^j}{j} 
\left[ \tfrac{1}{2}\left(\bar{S}_2+\bar{S}_1^2\right) 
-S_1 \bar{S}_1 
-\tfrac{1}{8}S_2 +\tfrac{1}{2}S_1^2 \right] \; .
\end{eqnarray}
All these equations are satisfied by analytic expressions
for these sums in terms of $\theta$ from Section~2 of this paper. 
Remember that one needs to substitute $\theta\to 2\theta$ 
as an argument on the r.h.s.

At order $\ep^3$, we have two independent structures,
$A_1^3$ and $A_1 A_2$. It is more convenient, however, 
to compare the coefficients of $A_1^3$ and $A_1(A_1^2-A_2)$.
The first equation coming from the coefficients of $A_1^3$ yields
\begin{equation}
\cot \tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j}
\left(
\tfrac{3}{16}C_0+\tfrac{3}{4}C_1+\tfrac{1}{2}C_2
\right)
=\tfrac{1}{2} 
\cot \theta
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}
\frac{{\widetilde{u}}^j}{j} 
\left(
\tfrac{1}{24}C_0 + \tfrac{1}{2}C_1 + \tfrac{1}{2}C_2
\right) \; , 
\end{equation}
where $C_j$ are the combinations of the harmonic sums
defined in Eqs.~(\ref{combinations2}).
This equation is also satisfied, if we use analytic expressions
for these sums in terms of $\theta$ given in Section~2.

The second equation at order $\ep^3$ comes from the
coefficient of $A_1(A_1^2-A_2)$,
\begin{equation}
\cot \tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j}
\left( -\tfrac{3}{4}S_1 S_2 + \tfrac{1}{2}S_2 \bar{S}_1
+\tfrac{1}{2}S_3 \right)
=\tfrac{1}{2} 
\cot \theta
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}
\frac{{\widetilde{u}}^j}{j} 
\left( -\tfrac{1}{8}S_1 S_2 + \tfrac{1}{8}S_2 \bar{S}_1
+\tfrac{1}{16}S_3 \right)\; .
\end{equation}
No explicit results for these sums are available. However,
if we introduce two functions 
\begin{eqnarray}
\Psi_1(\theta) &=& \cot\tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j}
\left(S_1S_2-S_3\right) \; ,
\\
\Psi_2(\theta) &=& \cot\tfrac{\theta}{2}
\sum_{j=1}^{\infty} \frac{1}{\left(2j\atop j\right)}\frac{u^j}{j}
\left(S_3-2S_2\bar{S}_1\right) \; ,
\end{eqnarray}
we obtain an interesting relation between them,
\begin{equation}
3\Psi_1(\theta) + \Psi_2(\theta) 
= \tfrac{1}{4}\Psi_1(2\theta) + \tfrac{1}{8}\Psi_2(2\theta) \; .
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The ${\cal O}(\alpha \alpha_{\rm s})$ corrections 
to the polarization function of  
neutral gauge bosons in arbitrary dimension}
\label{PoleMass}
\setcounter{equation}{0}
%=====================================================================
\begin{figure}[th]
\begin{center}
\centerline{\vbox{\epsfysize=30mm \epsfbox{gluon2.eps}}}
\caption{\label{QCD} 
Two-loop contributions to the off-shell polarization 
function of a neutral gauge boson.
Bold and thin lines correspond to the massive quark propagator
and the massless boson (gluon or photon) propagator, respectively.}
\end{center}
\end{figure}
%
%=====================================================================
Here we present an example of a physically relevant 
calculation\footnote{Another non-trivial example where 
these master integrals appear is given in Ref.~\cite{background}.}
that can be expressed in terms of 
the master integrals $J_{011}$ studied in Section~4.3.
Let us consider the two-loop propagator-type diagrams shown in Fig.~\ref{QCD}.
All of these 
${\cal O}(\alpha \alpha_{\rm s})$ 
contributions to the polarization function of
the gauge bosons 
involve a quark loop with a gluon exchange.
It was analytically calculated in~\cite{qcd},
up to the finite term of the $\ep$-expansion. 
Here we present the bare two-loop results in $n$-dimensional space-time
(see also in Ref.~\cite{poleII}). 
In contrast to the calculations performed in~\cite{qcd}, here we use 
Tarasov's recurrence relations~\cite{T97a} for the reduction of the 
original integrals to the set of master integrals.  
In this Appendix we use the Euclidean notation~\cite{diagramatic},
$P^2\leftrightarrow -p^2$, so that the on-shell limit
would read $P^2\to -m^2$.

Let us decompose the polarization tensor 
into the transverse $\Pi_{\rm T}(P^2)$ and longitudinal 
$\Pi_{\rm L}(P^2)$ parts,
\[
\Pi_{\mu \nu}(P^2) = \left( \delta_{\mu \nu} - \frac{P_\mu P_\nu}{P^2}\right) 
                    \Pi_{\rm T}(P^2) 
                  + \frac{P_\mu P_\nu }{P^2} \Pi_{\rm L}(P^2) \; . 
\]
Then, the two-loop corrections can be written as 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
&& \hspace*{-10mm}
\Pi_{\rm T}^{(2)}(P^2) =  \frac{g^2 g_{\rm s}^2}{(4 \pi)^{n/2}} N_c C_F    
\Biggl\{ 
- J_{011}(1,1,1) {\frac {4}{\left (n-1\right )t}}
\Bigl[
-4{ A}n{{ m}}^{2}
+12{ A}{{ m}}^{2}
+{ A}t{n}^{2}
\nonumber \\ && \hspace*{75mm}
-3{ A}tn
+2{ A}t
+St{n}^{2}
-3Stn
+2St
\Bigr]
%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && 
- J_{011}(1,1,2)
\frac {4}{\left (n-4\right )\left (n-3\right )\left (n-1\right )t}
\Bigl[
112{ A}{{ m}}^{4}n
+{ A}{t}^{2}{n}^{3}
-7{ A}{t}^{2}{n}^{2}
+18{ A}{t}^{2}n
\nonumber \\ && 
+S{t}^{2}{n}^{3}
-7S{t}^{2}{n}^{2}
+18S{t}^{2}n
-16{ A}{{ m}}^{4}{n}^{2}
-192{ A}{{ m}}^{4}
+4{ A}{{ m}}^{2}t{n}^{3}
-32{ A}{{ m}}^{2}t{n}^{2}
-16{ A}{t}^{2}
\nonumber \\ && 
+100{ A}{{ m}}^{2}tn
+4S{{ m}}^{2}t{n}^{3}
-36S{{ m}}^{2}t{n}^{2}
+96S{{ m}}^{2}tn
-112{ A}{{ m}}^{2}t
-16S{t}^{2}
-80S{{ m}}^{2}t
\Bigr]
%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && 
+ \left[ A_0(m) \right] ^2
{\frac {2 \left (n-2\right )}{\left (4{{ m}}^{2}+t\right )
       {{ m}}^{2}t\left (n-4\right )\left (n-1\right )\left (n-3\right )}}
\Bigl[
-4St{n}^{4}{{ m}}^{2}
+{ A}{t}^{2}{n}^{3}
+4{ A}{{ m}}^{2}t{n}^{3}
\nonumber \\ && 
+36S{{ m}}^{2}t{n}^{3}
+S{t}^{2}{n}^{3}
-32{ A}{{ m}}^{2}t{n}^{2}
-7S{t}^{2}{n}^{2}
-16{ A}{{ m}}^{4}{n}^{2}
-120S{{ m}}^{2}t{n}^{2}
-7{ A}{t}^{2}{n}^{2}
\nonumber \\ && 
+112{ A}{{ m}}^{4}n
+184S{{ m}}^{2}tn
+100{ A}{{ m}}^{2}tn
+18{ A}{t}^{2}n
+18S{t}^{2}n
-192{ A}{{ m}}^{4}
-16{ A}{t}^{2}
\nonumber \\ && 
-112S{{ m}}^{2}t
-112{ A}{{ m}}^{2}t
-16S{t}^{2}
\Bigr]
%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && 
- A_0(m) B_0(m,m,t) 
{\frac {2 \left (n-2\right )}{{{ m}}^{2}\left (n-3\right )
        \left (4{{ m}}^{2}+t\right )\left (n-4\right )\left (n-1\right )}}
\Bigl[
432{ A}{{ m}}^{4}n
-2{ A}t{n}^{4}{{ m}}^{2}
\nonumber \\ && 
+64{{ m}}^{4}S
+40{{ m}}^{4}Sn
+8{{ m}}^{4}S{n}^{3}
-48{{ m}}^{4}S{n}^{2}
+80{ A}{{ m}}^{4}{n}^{3}
-8{ A}{n}^{4}{{ m}}^{4}
-2St{n}^{4}{{ m}}^{2}
\nonumber \\ && 
+{ A}{t}^{2}{n}^{3}
-7{ A}{t}^{2}{n}^{2}
+18{ A}{t}^{2}n
+S{t}^{2}{n}^{3}
-7S{t}^{2}{n}^{2}
+18S{t}^{2}n
-280{ A}{{ m}}^{4}{n}^{2}
-256{ A}{{ m}}^{4}
\nonumber \\ && 
+24{ A}{{ m}}^{2}t{n}^{3}
-98{ A}{{ m}}^{2}t{n}^{2}
+180{ A}{{ m}}^{2}tn
+24S{{ m}}^{2}t{n}^{3}
-102S{{ m}}^{2}t{n}^{2}
+176S{{ m}}^{2}tn
\nonumber \\ && 
-16{ A}{t}^{2}
-128{ A}{{ m}}^{2}t
-16S{t}^{2}
-96S{{ m}}^{2}t
\Bigr]
%%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && 
- \left[ B_0(m,m,t) \right]^2
{\frac {2}{\left (4{{ m}}^{2}+t\right )\left (n-4\right )\left (n-1\right )}}
\Bigl[
448{ A}{{ m}}^{4}n
-32{{ m}}^{4}S
+16{ A}{{ m}}^{4}{n}^{3}
\nonumber \\ && 
+{ A}{t}^{2}{n}^{3}
-9{ A}{t}^{2}{n}^{2}
+30{ A}{t}^{2}n
+S{t}^{2}{n}^{3}
-9S{t}^{2}{n}^{2}
+30S{t}^{2}n
-144{ A}{{ m}}^{4}{n}^{2}
-448{ A}{{ m}}^{4}
\nonumber \\ && 
+8{ A}{{ m}}^{2}t{n}^{3}
-72{ A}{{ m}}^{2}t{n}^{2}
+232{ A}{{ m}}^{2}tn
+4S{{ m}}^{2}t{n}^{3}
-44S{{ m}}^{2}t{n}^{2}
+152S{{ m}}^{2}tn
\nonumber \\ && 
-32{ A}{t}^{2}
-240{ A}{{ m}}^{2}t
-32S{t}^{2}
-160S{{ m}}^{2}t
\Bigr] 
\Biggr\} \; ,
\label{PT}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\ && \hspace*{-10mm} 
\Pi_{\rm L}^{(2)}(P^2) =  A \frac{g^2 g_{\rm s}^2}{(4 \pi )^{n/2}} N_c C_F    
\Biggl\{ 
- J_{011}(1,1,1) \frac{ 16 m^{2}\left (n-3\right )}{t}
%%%%
\nonumber \\ && 
- J_{011}(1,1,2) \frac {16 m^{2}}{t\left (n-4\right )\left (n-3\right )}
\Bigl[
4   m^{2}{n}^{2}
-28 m^{2}n
+48 m^{2}
+t{n}^{2}-5tn+8t
\Bigr]
%%%%
\nonumber \\ && 
+ \left[ A_0(m) \right] ^2
\frac{8 \left (n-2\right )}{t\left ( 4 m^{2}+t \right )\left (n-4\right )}
\Bigl[
t{n}^{2}-3tn+4{{ m}}^{2}n-16{{ m}}^{2}
\Bigr]
%%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && 
- \left[ B_0(m,m,t) \right]^2
{\frac {8 {{ m}}^{2}}{\left (4{{ m}}^{2}+t\right )\left (n-4\right )}}
\Bigl[
-4tn+4t+4{{ m}}^{2}{n}^{2}-24{{ m}}^{2}n+t{n}^{2}+40{{m}}^{2}
\Bigr]
%%%%%%%%%%%%%%%%%%%%%
\nonumber \\ && 
- A_0(m) B_0(m,m,t) 
\frac{ 8 \left( n - 2 \right) \left( {n}^{2}-5n+8\right )}{ \left(4 m^{2}+t \right) (n-4) (n-3)}
\Bigl[-2{{ m}}^{2}n+10{{ m}}^{2}+t \Bigr] 
\Biggr\} \;, 
\label{PL}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the vertices $V_j$ ($j=1,2$) are defined as\footnote{Explicit 
values of coefficients  $v_j$ and $a_j$
for the Standard Model can be extracted from \cite{diagramatic}.}
$$
V_j = {\rm i} g \gamma_\mu (v_j + a_j \gamma_5) \;, 
$$
and we have introduced the following notations:
$$
A = a_1 a_2 \;, 
\quad 
S = v_1 v_2 \;, 
\quad
t=P^2,
$$
$N_c$ is a color factor (equal to 3 for quark and 1 for lepton), 
$C_F$ is the Casimir operator of the fundamental representation
of the Lie algebra (equal to $\tfrac{4}{3}$ for $SU(3)$ and 1 for QED), 
and $m$ is the mass of the loop fermion. Finally,
the occurring integrals are defined as
%
\begin{eqnarray}
J_{011}(\sigma, \nu_1, \nu_2) & = & 
\frac{\pi^{-n/2}}{\Gamma^2\left(3-\tfrac{n}{2}\right)}
\int \int \frac{{\rm d}^n K_1 \;\; {\rm d}^n K_2 }
{\left[(K_2-P)^2\right]^{\sigma}
\left[ K_1^2+m^2 \right]^{\nu_1}
\left[ (K_1-K_2)^2 + m^2 \right]^{\nu_2}} \; ,
\nonumber \\ 
B_0(m_a,m_b,t) & = & \frac{\pi^{-n/2}}{\Gamma\left(3-\tfrac{n}{2}\right)}
\int \frac{{\rm d}^n K}
{\left[(K-P)^2+m_a^2\right]
\left[ K^2+m_b^2 \right]} \; ,
\nonumber \\ 
A_0(m) & = & \frac{\pi^{-n/2}}{\Gamma\left(3-\tfrac{n}{2}\right)}
\int \frac{{\rm d}^n K}{K^2+m^2}
\equiv \frac{4 (m^2)^{n-2}}{(n-2)(n-4)} \;.
\end{eqnarray}
We note that the integral $J_{011}$ is defined in the same way
as in Section~4.3, one should only remember to substitute $p^2\to-t$.
 
In particular, for the zero momentum transfer, $P^2=t=0$, we get
\begin{eqnarray}
\Pi_{\rm L}^{(2)}(0) & = & 
- A \frac{g^2 g_{\rm s}^2}{(4 \pi)^{n/2}} N_c  C_F \left[ A_0(m) \right]^2 
\frac{4(n-2)(n^2-5n+7)}{m^2} \;, 
\nonumber \\
\Pi_{\rm T}^{(2)}(0) & = & 
- A \frac{g^2 g_s^2}{(4 \pi)^{n/2}} N_c  C_F \left[ A_0(m) \right]^2 
\frac{4(n-2)(n^3-6n^2+13n-11)}{(n-1)m^2} \;. 
\nonumber 
\end{eqnarray}

To obtain the finite terms of the $\ep$-expansion of the results
given in Eqs.~(\ref{PT}) and (\ref{PL}),
the integral $J_{011}(1,1,2)$ should be expanded up to the $\ep$-part.
Furthermore,
using the approach of Ref.~\cite{T97a}, we obtain the following 
relations between the integrals investigated in 
Section~4.3 and the master integrals $J_{011}(1,1,1)$ and  $J_{011}(1,1,2)$:
\begin{eqnarray}
&& 
\hspace*{-10mm}
J_{011}(1,1,1)\frac{(3n-8)(3n-10)(n-3)^2}{n-4}  = 
\left[ A_0(m)  \right]^2 (n-3)(n-2)^2 
\Biggl[
\frac{t}{4m^4} 
- \frac{(7n-24)}{2m^2 (n-4)}
\Biggr]
\nonumber \\ && 
+J_{011}(1,2,2)
\Biggl[ \frac{8 m^2 (t+m^2)}{n-4}
- 2 t^2 (n-3) + 2 t m^2 (7n-17) + 8 m^4 (2n-5)
\Biggr]
\nonumber \\ && 
+ \left[ J_{011}(1,2,2) + 2 J_{011}(2,1,2) \right] (n-3) (t+4m^2) (t-2 m^2)
\;, 
\\ 
&& 
\hspace*{-10mm}
J_{011}(1,1,2)\frac{(3n-8)(3n-10)(n-3)^2}{n-4}  =
\left[ A_0(m)  \right]^2 
\frac{(n-3)(2n-7)(3n-8)(n-2)^2}{4m^4 (n-4)}
\nonumber \\ && 
+J_{011}(1,2,2)
\Biggl[
-\frac{ 4 (t+2m^2)}{n-4}
- n^2 \left( \tfrac{15}{2}t + 12m^2 \right) + n \left( \tfrac{79}{2}t+62m^2 \right) - 55 t - 86 m^2
\Biggr]
\nonumber \\ && 
+ \tfrac{1}{2} \left[ J_{011}(1,2,2) + 2 J_{011}(2,1,2) \right] (n-3) (3n-8)(t+4m^2) 
\;. 
\end{eqnarray}
%
From these relations we see that, in order to obtain the $\ep^j$ terms of 
$J_{011}(1,1,1)$ and  $J_{011}(1,1,2)$, the integral
$J_{011}(1,2,2)$ should be expanded up to $\ep^j$, whereas
the combination $[J_{011}(1,2,2)+2J_{011}(2,1,2)]$ 
up to $\ep^{j-1}$ only.
The expansion of the integrals near the threshold can be performed by using 
the numerical algorithm described in Ref.~\cite{expansion}.

The higher-order terms of the $\ep$-expansion 
of the one-loop integral $B_0(m_1,m_2,t)$ 
can be extracted from Refs.~\cite{D-ep,DK01}. 
Here we present coefficients up to the order $\ep^2$ 
for the particular case $m_1=m_2=m$:
\begin{eqnarray}
\hspace{-5mm}
B_0(m,m,t) & \!\!=\!\! & \frac{(m^2)^{-\ep}}{(1-2\ep) }
\Biggl(
\frac{1}{\ep} + \frac{1+y}{1-y} \Biggl\{ \ln y 
+ \ep \Bigl[  
\tfrac{1}{2} \ln^2 y \!-\! 2 \ln y \ln (1+y) \!-\! 2 \Li{2}{-y} 
\!-\! \zeta_2 
       \Bigr] 
\nonumber \\ && 
+ \ep^2 \Bigl[ 
4 \Snp{1,2}{-y} 
- 2 \Li{3}{-y} 
+ 4 \ln(1+y) \Li{2}{-y} 
- \ln^2 y \ln(1+y) 
\nonumber \\ && 
+ 2 \ln y \ln^2 (1+y) 
+ \tfrac{1}{6} \ln^3 y 
- \zeta_2 \ln y 
+ 2 \zeta_2 \ln(1+y) 
- 2\zeta_3 
\Bigr]  
+ {\cal O}(\ep^3)
\Biggr\}
\Biggr) \;, 
\hspace*{6mm}
\end{eqnarray}
where the variable $y$ is defined in Section~3.

The bare two-loop amplitudes (\ref{PT}) and (\ref{PL}) contain 
subdivergencies which should be cancelled by proper counterterms,
$$
\Pi_{\rm CT}(P^2) = 
\delta m^2 \frac{\partial}{\partial m^2 } \Pi^{(1)} (P^2) \;, 
$$
where $\Pi^{(1)}$ is the one-loop amplitude, while $\delta m^2 $ 
is the one-loop mass
counterterm defined in a particular renormalization scheme. 
The derivatives of the bare one-loop amplitudes in $n$ dimensions
read
\begin{eqnarray}
\frac{\partial}{\partial m^2} \Pi_{\rm T}^{(1)}(P^2) 
& \!\!=\!\! & \frac{g^2}{(4 \pi )^{n/2}} N_c 4
\Biggl\{ 
\Biggl[ \frac{S t (3\!-\!n) }{(4m^2\!+\!t)} \!+\! S\!+\!(2\!-\!n)A \Biggr]B_0(m,m,t)
\!+\! \frac{2 S (n\!-\!2) }{(4m^2\!+\!t)} A_0(m)
\Biggr \} \;, 
\nonumber \\
\frac{\partial}{\partial m^2 } \Pi_{\rm L}^{(1)}(P^2) & \!\!=\!\! & 
A \frac{g^2}{(4 \pi )^{n/2}} N_c 4 
\Biggl \{ 
\Biggl[ \frac{t (n-3) }{(4m^2+t)} + 1-n \Biggr]B_0(m,m,t)
+ \frac{2 (2-n) }{(4m^2+t)} A_0(m)
\Biggr \} \;.
\nonumber 
\end{eqnarray}

As an example of application of these formulae, let us consider 
the transversal part of the $\gamma-Z$ propagator $(A=0)$
in the $\overline{MS}$-scheme.
Up to the finite in $\ep$ part, the result for the subtracted quantity 
$\Pi^{\rm sub}(P^2) = \Pi^{(2)}(P^2) + \Pi_{\rm CT}$ is 
\begin{eqnarray}
\label{gammaZ}
\Pi_{{\rm T},\gamma Z}^{\rm sub}(P^2) 
&=& S  \frac{g^2 g_{\rm s}^2}{(4 \pi)^2} N_c C_F    
\Biggl\{ 
- \frac{2t}{\ep} 
+ \left(-\tfrac{55}{3}t + \tfrac{296}{3} m^2 \right)
+ 16 t \zeta_3 \left[ 1 - 4 \frac{y^2}{(1-y)^4} \right]
\nonumber \\ && 
- \tfrac{16}{3} m^2 \frac{(1-4y+y^2)}{y(1-y)^2} 
\left [ \ln(1-y) + 2 \ln(1+y) \right] \ln y 
\left[ (1+y^2) \ln y - 2 (1-y^2) \right]
\nonumber \\ && 
+ \tfrac{8}{3} m^2 \frac{(2+7y-22y^2+6y^3)}{(1-y)^2} \ln^2 y
- 4 m^2 \ln y \frac{(1-6y-46y^2-6y^3+y^4)}{y(1-y^2)}
\nonumber \\ && 
+ 4 t \ln \frac{m^2}{\mu^2} \left[ 1- \frac{12 y}{(1-y)^2} \right] 
- 96 m^2 \frac{y}{1-y^2} \ln y \ln \frac{m^2}{\mu^2}
\nonumber \\ && 
+ \tfrac{32}{3} m^2 \frac{(1-4y+y^2)}{y(1-y)^2} 
\left [ \Li{2}{y} + 2 \Li{2}{-y} \right]  
\left[ 1-y^2 - 2\left(1+y^2\right) \ln y \right] 
\nonumber \\ && 
+ 32 m^2 \frac{(1-4y+y^2)}{y(1-y)^2} (1+y^2) 
\left [\Li{3}{y} + 2 \Li{3}{-y} \right]  
+ {\cal O}(\ep)
\Biggr\} \; , 
\end{eqnarray}
%
where we have taken into account the one-loop massive counterterm
$$
\delta m^2 =  - 2 \frac{g_{\rm s}^2}{(4\pi)^2} C_F \frac{3}{\ep} m^2 \;. 
$$
The small-momentum expansion of Eq.~(\ref{gammaZ}) is 
\begin{eqnarray}
\hspace{-10mm}
&& 
\left. \Pi_{T,\gamma Z}^{sub}(p^2) \right|_{P^2 \to 0 }= 
S  \frac{g^2 g_{\rm s}^2}{(4 \pi)^2} m^2 N_c C_F  
\Biggl\{ 
\frac{2}{\ep} u
+ \tfrac{13}{3} u + \tfrac{776}{405} u^2 
+ \ln \frac{m^2}{\mu^2 } \left( 4 u + \tfrac{8}{5} u^2 \right)
+ {\cal O}(u^3)
\Biggr\} \;, 
\nonumber 
\end{eqnarray}
where $u=-P^2/m^2$.
%
Let us remind that in a theory with the spontaneous symmetry breaking, 
like the Standard Model, 
the inclusion of tadpoles~\cite{FJ} is also required 
for the renormalization group 
invariance of the massive parameters~\cite{poleI,poleII,expansion}.
The proper bare two-loop tadpole contribution 
is given in Section~4.3 of Ref.~\cite{poleII}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
\bibitem{dimreg}
G.~'tHooft and M.~Veltman,
Nucl.\ Phys.\ {\bf B44} (1972) 189;\\
%%CITATION = NUPHA,B44,189;%%
C.G.~Bollini and J.J.~Giambiagi,
Nuovo~Cimento {\bf 12B} (1972) 20; \\
%%CITATION = NUCIA,B12,20;%%
J.F.~Ashmore,  
Lett.\ Nuovo Cim.\ {\bf 4} (1972) 289;\\
%%CITATION = NCLTA,4,289;%%
G.M.~Cicuta and E.~Montaldi,
Lett.\ Nuovo Cim.\ {\bf 4} (1972) 329.
%%CITATION = NCLTA,4,329;%%

\bibitem{Lewin}
L.~Lewin, {\it Polylogarithms and associated functions}
(North-Holland, Amsterdam, 1981).

\bibitem{Nielsen}
K.S.~K\"olbig, J.A.~Mignaco and E.~Remiddi, B.I.T. {\bf 10} (1970) 38;\\
%%CITATION = BITXA,10,38;%%
R.~Barbieri, J.A.~Mignaco and E.~Remiddi, Nuovo Cim.\ {\bf A11} (1972) 824;\\
%%CITATION = NUCIA,A11,824;%%
A.~Devoto and D.W.~Duke, Riv.\ Nuovo Cim.\  {\bf 7}, No.6 (1984) 1;\\
%%CITATION = RNCIB,7N6,1;%%
K.S.~K\"olbig, SIAM J.\ Math.\ Anal.\ {\bf 17} (1986) 1232.
%%CITATION = SJMAA,17,1232;%%

\bibitem{RV00}
E.~Remiddi and J.A.M.~Vermaseren,
Int.\ J.\ Mod.\ Phys.\ {\bf A15} (2000) 725.
%%CITATION = IMPAE,A15,725;%%
 
\bibitem{GR2001}
A.B.~Goncharov,
Math.\ Res.\ Lett.\ {\bf 5} (1998) 497; \\
%%CITATION = 00146,5,497;%%
J.M.~Borwein, D.M.~Bradley, D.J.~Broadhurst and P.~Lison\v ek,
Trans.\ Amer.\ Math.\ Soc.\ {\bf 353} (2001) 907; \\
%%CITATION = TAMTA,353,907;%%
T.~Gehrmann and E.~Remiddi,
Nucl.\ Phys.\ {\bf B601} (2001) 248.
%%CITATION = NUPHA,B601,248;%%

\bibitem{nested}
S.~Moch, P.~Uwer and S.~Weinzierl,
J.\ Math.\ Phys.\  {\bf 43} (2002) 3363.
%%CITATION = HEP-PH 0110083;%%

\bibitem{poleII}
F.~Jegerlehner, M.Yu.~Kalmykov and O.~Veretin, 
to be publ. in Nucl.\ Phys.\ {\bf B}
(hep-ph/0212319).
%%CITATION = HEP-PH 0212319;%%

\bibitem{dem}
A.V.~Kotikov,
Phys.\ Lett.\ {\bf B254} (1991) 158;
%%CITATION = PHLTA,B254,158;%%
{\bf B259} (1991) 314. 
%%CITATION = PHLTA,B259,314;%%

\bibitem{BD-TMF}
E.E.~Boos and A.I.~Davydychev,
Teor.\ Mat.\ Fiz.\ {\bf 89} (1991) 56.
%%CITATION = TMFZA,89,56;%%

\bibitem{FKV98}
J.~Fleischer, A.V.~Kotikov and O.L.~Veretin,
Phys.\ Lett.\ {\bf B417} (1998) 163.
%%CITATION = HEP-PH 9707492;%%

\bibitem{DD} A.I.~Davydychev and R.~Delbourgo, 
J.\ Math.\ Phys.\ {\bf 39} (1998) 4299. 
%%CITATION = JMAPA,39,4299;%%

\bibitem{FKV99}
J.~Fleischer, A.V.~Kotikov and O.L.~Veretin,
Nucl.\ Phys.\ {\bf B547} (1999) 343.
%%CITATION = NUPHA,B547,343;%%

\bibitem{DK-bastei} 
A.I.~Davydychev and M.Yu.~Kalmykov,
Nucl.\ Phys.\ B (Proc.\ Suppl.) {\bf 89} (2000) 283.
%%CITATION = HEP-TH 0005287;%%

\bibitem{DK01}
A.I.~Davydychev and  M.Yu.~Kalmykov,
Nucl. Phys. {\bf B605} (2001) 266.
%%CITATION = HEP-TH 0012189;%%

\bibitem{oneloop}
J.~Fleischer, F.~Jegerlehner and O.V.~Tarasov, 
DESY preprint DESY-03-033.

\bibitem{single}
J.~Fleischer, M.Yu.~Kalmykov and  A.V.~Kotikov,
Phys.\ Lett.\ {\bf B462} (1999) 169; {\bf B467} (1999) 310(E).
%%CITATION = PHLTA,B462,169;%%

\bibitem{harmonic}
D.I.~Kazakov and A.V.~Kotikov, 
Theor.\ Math.\ Phys.\  {\bf 73} (1988) 1264; 
%%CITATION = TMPHA,73,1264;%% 
Nucl.\ Phys.\ {\bf B307} (1988) 721; \\
%%CITATION = NUPHA,B307,721;%% 
D.~Borwein, J.M.~Borwein and R.~Girgensohn,
Proc.\ Edinburgh Math.\ Soc.\ {\bf 38} (1995) 277;\\
%%CITATION = PEMSA,38,277;%%
J.M.~Borwein and R.~Girgensohn,
Electronic J.\ Combinatorics {\bf 3} (1996) R23;\\
%%CITATION = 00185,3,R23;%%
P.~Flajolet and B.~Salvy, Experimental Math.\ {\bf 7} (1998) 15;\\
%%CITATION = 00162,7,15;%%
O.M.~Ogreid and P.~Osland, J.\ Comput.\ Appl.\ Math.\ {\bf 98} (1998) 245;
%%CITATION = JCAMD,98,245;%%
{\bf 136} (2001) 389; \\
%%CITATION = HEP-TH 9904206;%%
V.A.~Smirnov,
Phys.\ Lett.\ {\bf B547} (2002) 239. 
%%CITATION = HEP-PH 0209193;%%

\bibitem{KV00}
M.Yu.~Kalmykov and O.~Veretin, Phys.\ Lett.\ {\bf B483} (2000) 315.
%%CITATION = PHLTA,B483,315;%%

\bibitem{odd}
J.~Fleischer and M.Yu.~Kalmykov,
%``Single mass scale diagrams: Construction of a basis for the  epsilon-expansion,''
Phys.\ Lett.\ {\bf B470} (1999) 168.
%%CITATION = HEP-PH 9910223;%%

\bibitem{euler}
D.J.~Broadhurst, 
Open University preprint OUT-4102-62 (hep-th/9604128).
%%CITATION = HEP-TH 9604128;%%

\bibitem{B99}
D.J.~Broadhurst, Eur.\ Phys.\ J.\ {\bf C8} (1999) 311.
%%CITATION = EPHJA,C8,311;%%

\bibitem{review}
A.I.~Davydychev and M.Yu.~Kalmykov,
%``Geometrical approach to loop calculations and the epsilon-expansion of  Feynman diagrams,''
Proc. Workshop ``CPP2001'', Tokyo, Japan, November 2001, 
KEK Proceedings 2002-11, p.~169
(hep-th/0203212).
%%CITATION = HEP-TH 0203212;%%

\bibitem{sums}
Z.~Nan-Yue and K.S.~Williams, Pacific J.\ Math.\ {\bf 168} (1995) 271.
%%CITATION = PJMAA,168,271;%%

\bibitem{BBK}
J.M.~Borwein, D.J.~Broadhurst and J.~Kamnitzer,
Experimental Math. {\bf 10} (2001) 25.
%%CITATION = HEP-TH 0004153;%%

\bibitem{Ls_ex}
A.I.~Davydychev and J.B.~Tausk,
Phys.\ Rev.\ {\bf D53} (1996) 7381; \\
%%CITATION = PHRVA,D53,7381;%%
A.I.~Davydychev,
%Mainz preprint MZ-TH/99-30
Proc.\ Workshop ``AIHENP-99'', Heraklion, Greece, April 1999 (Parisianou S.A.,
Athens, 2000), p.~219
(hep-th/9908032);\\
%%CITATION = HEP-TH 9908032;%% 
A.V.~Kotikov and L.N.~Lipatov,
Nucl.\ Phys.\ {\bf B582} (2000) 19.
%%CITATION = NUPHA,B582,19;%%

\bibitem{D-ep}
A.I.~Davydychev, Phys.\ Rev.\ {\bf D61} (2000) 087701.
%%CITATION = PHRVA,D61,087701;%%

\bibitem{PBM3} A.P.~Prudnikov,  Yu.A.~Brychkov  and O.I.~Marichev,
{\em Integrals and Series, v.3: More Special Functions},
Gordon and Breach, New York, 1990.

\bibitem{BFT93}
D.J.~Broadhurst, J.~Fleischer and O.V.~Tarasov,
Z.\ Phys.\ {\bf C60} (1993) 287.
%%CITATION = ZEPYA,C60,287;%%

\bibitem{B90}
D.J.~Broadhurst, Z.\ Phys.\ {\bf C47} (1990) 115.
%%CITATION = ZEPYA,C47,115;%%

\bibitem{D91}
A.I.~Davydychev,
{\em ``Loop calculations in QCD with massive quarks"},
talk at Int.\ Conf.\ ``Relativistic Nuclear Dynamics"
(Vladivostok, Russia, September 1991),\\
{\tt http://wwwthep.physik.uni-mainz.de/$\;\widetilde{}\;$davyd/preprints/vladiv.ps.gz}

\bibitem{T97a}
O.V.~Tarasov, Nucl.\ Phys.\ {\bf B502} (1997) 455.
%%CITATION = NUPHA,B502,455;%%

\bibitem{thresholds}
A.I.~Davydychev and V.A.~Smirnov, Nucl.\ Phys.\ {\bf B554} (1999) 391.
%%CITATION = NUPHA,B554,391;%%

\bibitem{T96}
O.V.~Tarasov, Phys.\ Rev.\ {\bf D54} (1996) 6479.
%%CITATION = HEP-TH 9606018;%%

\bibitem{Hgg}
A.~Djouadi, M.~Spira and P.M.~Zerwas,
%``Production of Higgs bosons in proton colliders: QCD corrections,''
Phys.\ Lett.\ {\bf B264} (1991) 440; \\
%%CITATION = PHLTA,B264,440;%%
S.~Dawson and R.~Kauffman,
%``QCD corrections to Higgs boson production: nonleading terms in the heavy quark limit,''
Phys.\ Rev.\ {\bf D49} (1994) 2298; \\
%%CITATION = HEP-PH 9310281;%%
M.~Spira, A.~Djouadi, D.~Graudenz and P.M.~Zerwas,
%``Higgs boson production at the LHC,''
Nucl.\ Phys.\ {\bf B453} (1995) 17.
%%CITATION = HEP-PH 9504378;%%

\bibitem{vertex}
A.I.~Davydychev and V.A.~Smirnov, 
to be publ. in Nucl.\ Instr.\ Meth.\ {\bf A} 
%``Analytical evaluation of certain on-shell two-loop three-point  diagrams,''
(hep-ph/0210171); \\
%%CITATION = HEP-PH 0210171;%%
R.~Bonciani, P.~Mastrolia and E.~Remiddi,
%``Vertex diagrams for the QED form factors at the 2-loop level,''
hep-ph/0301170.
%%CITATION = HEP-PH 0301170;%%

\bibitem{GKP}
S.~Groote, J.~G.~Korner and A.~A.~Pivovarov,
%``Transcendental numbers and the topology of three-loop bubbles,''
Phys.\ Rev.\ {\bf D60} (1999) 061701.
%%CITATION = HEP-PH 9904304;%%

\bibitem{PSLQ} 
H.R.P.~Ferguson, D.H.~Bailey and S. Arno, 
Math.\ Comput.\ {\bf 68} (1999) 351.
%%CITATION = MCMPA,68,351;%% 

\bibitem{background}
F.~Jegerlehner and O.V.~Tarasov, Nucl.\ Phys.\ {\bf B549} (1999) 481.
%%CITATION = HEP-PH 9809485;%%

\bibitem{qcd}
T.H.~Chang, K.J.~Gaemers and W.L.~van Neerven, Nucl.\ Phys.\ {\bf B202} (1982) 407; \\
%%CITATION = NUPHA,B202,407;%%
A.~Djouadi and C.~Verzegnassi, Phys.\ Lett.\ {\bf B195} (1987) 265; \\
%%CITATION = PHLTA,B195,265;%%
A.~Djouadi, Nuovo Cim.\ {\bf A100} (1988) 357; \\
%%CITATION = NUCIA,A100,357;%%
B.A.~Kniehl, J.H.~K\"uhn and R.G.~Stuart, Phys.\ Lett.\ {\bf B214} (1988) 621;\\
%%CITATION = PHLTA,B214,621;%%
B.A.~Kniehl, Nucl.\ Phys.\ {\bf B347} (1990) 86 ; \\
%%CITATION = NUPHA,B347,86;%%
A.~Djouadi and P.~Gambino, Phys.\ Rev.\ {\bf D49} (1994) 3499;
{\bf D53} (1996)  4111(E).
%%CITATION = HEP-PH 9309298;%%

\bibitem{diagramatic}
M.~Veltman, {\it Diagrammatica}, Cambridge University Press, 1994; \\
D.~Bardin and G.~Passarino, {\it The Standard Model in the Making}, Oxford, UK: Clarendon, 1999.

\bibitem{expansion}
F.~Jegerlehner, M.Yu.~Kalmykov and O.~Veretin, 
Nucl.\ Phys.\ B (Proc.\ Suppl.) {\bf 116} (2003) 382 (hep-ph/0212003).
%%CITATION = HEP-PH 0212003;%%

\bibitem{FJ} 
J.~Fleischer and F.~Jegerlehner, 
Phys.\ Rev.\ {\bf D23} (1981) 2001. 
%%CITATION = PHRVA,D23,2001;%%

\bibitem{poleI}
F.~Jegerlehner, M.Yu.~Kalmykov and O.~Veretin, 
Nucl.\ Phys.\ {\bf B641} (2002) 285.
%%CITATION = HEP-PH 0105304;%%

\bibitem{S_Bha}
V.A.~Smirnov,
Phys.\ Lett.\ {\bf B524} (2002) 129.
%%CITATION = PHLTA,B524,129;%% 

\bibitem{d_3}
K.G.~Chetyrkin and M.~Steinhauser,
Nucl.\ Phys.\ {\bf B573} (2000) 617.
%%CITATION = HEP-PH 9911434;%%

\bibitem{leo96}
L.V.~Avdeev, Comput.\ Phys.\ Commun.\ {\bf 98} (1996) 15.
%%CITATION = CPHCB,98,15;%%

\bibitem{matad}
M.~Steinhauser, Comput.\ Phys.\ Commun.\ {\bf 134} (2001) 335.
%%CITATION = HEP-PH 0009029;%%


\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}


