\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}

\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}

\newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}}

\begin{document}

\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}

\begin{center}
\vskip 1cm{\LARGE\bf 
Vacca-Type  Series for Values of the\\
\vskip 0in Generalized Euler Constant Function \\
\vskip .13in
and its Derivative
}
\vskip 1cm
\large
Khodabakhsh Hessami Pilehrood\footnote{Author's current address:
Mathematics Department, Faculty of Basic Sciences, Shahrekord
University, Shahrekord, P.O. Box 115, Iran.  Research was in part
supported by a grant from IPM (No.\ 87110018).}
and
Tatiana Hessami Pilehrood\footnote{Author's current address:
Mathematics Department, Faculty of Basic Sciences, Shahrekord
University, Shahrekord, P.O. Box 115, Iran.  Research was in part
supported by a grant from IPM (No.\ 87110019).} \\
Institute for Studies in Theoretical Physics and Mathematics \\
Tehran, Iran \\
\href{mailto:hessamik@ipm.ir}{\tt hessamik@ipm.ir}\\
\href{mailto:hessamik@gmail.com}{\tt hessamik@gmail.com} \\
\href{mailto:hessamit@ipm.ir}{\tt hessamit@ipm.ir} \\
\href{mailto:hessamit@gmail.com}{\tt hessamit@gmail.com} 
\end{center}

\vskip .2 in

\begin{abstract}
We generalize well-known Catalan-type integrals for Euler's
constant to values of the ge\-ne\-ra\-li\-zed Euler constant
function and its derivatives.
 Using generating functions appearing in
these integral representations, we give new Vacca and
Ramanujan-type series for values of the generalized Euler constant
function and Ad\-dison-type series for values of the
generalized Euler constant function and its derivative. As a
consequence, we get base-$B$ rational series for
$\log\frac{4}{\pi},$ $\frac{G}{\pi}$ (where $G$ is Catalan's
constant), $\frac{\zeta'(2)}{\pi^2}$, and also for logarithms of
the Somos and Glaisher-Kinkelin constants.
\end{abstract}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}
\label{intro}

J.~Sondow \cite{s} proved the following two formulas:
\begin{equation}
\qquad\gamma=\sum_{n=1}^{\infty}\frac{N_{1,2}(n)+N_{0,2}(n)}{2n(2n+1)},
\label{eq1}
\end{equation}
\begin{equation}
\log\frac{4}{\pi}=\sum_{n=1}^{\infty}\frac{N_{1,2}(n)-N_{0,2}(n)}{2n(2n+1)},
\label{eq2}
\end{equation}
where $\gamma$ is Euler's constant and $N_{i,2}(n)$ is the number
of $i$'s in the binary expansion of $n$ (see sequences \seqnum{A000120} and
\seqnum{A023416} in Sloane's database \cite{oeis}). 
The series (\ref{eq1}) is
equivalent to  the well-known Vacca series \cite{va}
\begin{equation}
\gamma=\sum_{n=1}^{\infty}(-1)^n\frac{\lfloor\log_2n\rfloor}{n}=\sum_{n=1}^{\infty}
(-1)^n\frac{N_{1,2}\bigl(\lfloor\frac{n}{2}\rfloor\bigr)+
N_{0,2}\bigl(\lfloor\frac{n}{2}\rfloor\bigr)}{n} \label{eq3}
\end{equation}
and both series (\ref{eq1}) and (\ref{eq3}) may be derived from
Catalan's integral \cite{ca}
\begin{equation}
\gamma=\int_0^1\frac{1}{1+x}\sum_{n=1}^{\infty}x^{2^n-1}\,dx.
\label{eq4}
\end{equation}
To see this it  suffices to note that
$$
G(x)=\frac{1}{1-x}\sum_{n=0}^{\infty}x^{2^n}=\sum_{n=1}^{\infty}
(N_{1,2}(n)+N_{0,2}(n))x^n
$$
is a generating function of the sequence $N_{1,2}(n)+N_{0,2}(n),$
(see \seqnum{A070939}), which is the binary length of
$n,$ rewrite (\ref{eq4}) as
$$
\gamma=\int_0^1(1-x)\frac{G(x^2)}{x}\,dx
$$
and integrate the power series termwise. In view of the equality
\begin{equation*}
1=\int_0^1\sum_{n=1}^{\infty}x^{2^n-1}\,dx,
\end{equation*}
which is  easily verified by termwise integration, (\ref{eq4}) is
equivalent to the formula
\begin{equation}
\gamma=1-\int_0^1\frac{1}{1+x}\sum_{n=1}^{\infty}x^{2^n}\,dx
\label{eq05}
\end{equation}
obtained independently by Ramanujan (see \cite[Corollary 2.3]{bebo}).
Catalan's integral (\ref{eq05}) gives the following rational
series for $\gamma:$
\begin{equation}
\gamma=1-\int_0^1(1-x)G(x^2)\,dx=1-\sum_{n=1}^{\infty}
\frac{N_{1,2}(n)+N_{0,2}(n)}{(2n+1)(2n+2)}. \label{eq06}
\end{equation}
Averaging (\ref{eq1}), (\ref{eq06}) and (\ref{eq4}), (\ref{eq05}),
respectively, we get Addison's series for $\gamma$ \cite{ad}
$$
\gamma=\frac{1}{2}+\sum_{n=1}^{\infty}\frac{N_{1,2}(n)+N_{0,2}(n)}%
{2n(2n+1)(2n+2)}
$$
and its corresponding integral
\begin{equation}
\gamma=\frac{1}{2}+\frac{1}{2}\int_0^1\frac{1-x}{1+x}\sum_{n=1}^{\infty}
x^{2^n-1}\,dx, \label{eq065}
\end{equation}
 respectively. Integrals (\ref{eq05}),
(\ref{eq4}) were generalized to an arbitrary integer base $B>1$ by
S.~Ramanujan and by  B.~C.~Berndt and D.~C.~Bowman (see \cite{bebo}):
\begin{equation}
\gamma
=1-\int_0^1\left(\frac{1}{1-x}-\frac{Bx^{B-1}}{1-x^B}\right)
\sum_{n=1}^{\infty}x^{B^n}\,dx \qquad\qquad (\text{Ramanujan}),
\label{eq07}
\end{equation}
 \begin{equation}
  \gamma
=\int_0^1\left(\frac{B}{1-x^B}-\frac{1}{1-x}\right)\sum_{n=1}^{\infty}
x^{B^n-1}\,dx \qquad\qquad (\text{Berndt-Bowman}). \label{eq08}
\end{equation}
Formula (\ref{eq08}) implies the generalized Vacca series for
$\gamma$ (see \cite[Theorem 2.6]{bebo}) proposed by L.~Carlitz
\cite{carl}:
\begin{equation}
\gamma=\sum_{n=1}^{\infty}\frac{\varepsilon(n)}{n}\lfloor\log_Bn\rfloor,
\label{eq09}
\end{equation}
 where
\begin{equation}
\varepsilon(n)=\begin{cases}
B-1,     &  \text{if $B$ divides $n$}; \\
 -1, & \text{otherwise};
 \end{cases}
 \label{eq095}
\end{equation}
and the averaging integral of (\ref{eq07}) and (\ref{eq08})
produces the generalized Addison series for $\gamma$ found by
Sondow   \cite{s}:
\begin{equation}
\gamma=\frac{1}{2}+\sum_{n=1}^{\infty}\frac{\lfloor\log_BBn\rfloor
P_B(n)}{Bn(Bn+1)\cdots (Bn+B)}, \label{eq010}
\end{equation}
 where $P_B(x)$ is a
polynomial of degree $B-2$ defined  by
\begin{equation}
P_B(x)=(Bx+1)(Bx+2)\cdots
(Bx+B-1)\sum_{m=1}^{B-1}\frac{m(B-m)}{Bx+m}. \label{polyn}
\end{equation}
 In this
 paper, we consider the generalized Euler constant
function
\begin{equation}
\gamma_{a,b}(z)=\sum_{n=0}^{\infty}\left(\frac{1}{an+b}-
\log\left(\frac{an+b+1}{an+b}\right)\right)z^n, \qquad a,b
\in{\mathbb N}, \qquad |z|\le 1, \label{eq7}
\end{equation}
which is related to the constants in (\ref{eq1}),
(\ref{eq2}) as $\gamma_{1,1}(1)=\gamma,$
$\gamma_{1,1}(-1)=\log\frac{4}{\pi}.$
Basic properties of a special case of this function, $\gamma_{1,1}(z),$
were studied earlier in \cite{sh,he0}.
In Section~\ref{S2}, we show that $\gamma_{a,b}(z)$ admits an analytic
continuation to the domain ${\mathbb C}\setminus [1,+\infty)$
in terms of the Lerch transcendent.  In Sections~\ref{S3}--\ref{S4},
we generalize Catalan-type integrals (\ref{eq07}),
(\ref{eq08}) to values of the generalized Euler constant
function
and its derivatives.
 Using generating functions appearing in
these integral representations, we give new Vacca- and
Ramanujan-type series for values of $\gamma_{a,b}(z)$ and
Addison-type series for values of $\gamma_{a,b}(z)$ and its
derivative. In Section \ref{S5}, we get base-$B$ rational series for
$\log\frac{4}{\pi},$ $\frac{G}{\pi},$ (where $G$ is Catalan's
constant), $\frac{\zeta'(2)}{\pi^2}$ and also for logarithms of the
Somos and Glaisher-Kinkelin constants. We also mention a
connection of our approach to summation of series of the form
$$\
\sum_{n=1}^{\infty}N_{\omega,B}(n)Q(n,B) \quad\text{and}\quad
\sum_{n=1}^{\infty}\frac{N_{\omega,B}(n)P_B(n)}{Bn(Bn+1)\cdots
(Bn+B)},
$$
where $Q(n,B)$ is a rational function of $B$ and $n$
\begin{equation}
Q(n,B)=\frac{1}{Bn(Bn+1)}+\frac{2}{Bn(Bn+2)}+\cdots+\frac{B-1}{Bn(Bn+B-1)},
\label{eq5}
\end{equation}
and $N_{\omega,B}(n)$ is the number of occurrences of a word
$\omega$ over the alphabet $\{0,1,\ldots, B-1\}$ in the $B$-ary
expansion of  $n,$ considered in \cite{assh}.
Moreover, we answer some  questions posed in \cite{assh} concerning
possible generalizations of the series (\ref{eq1}) and (\ref{eq2}) to any integer base $B>1.$
Note that in the above notation,
the generalized Vacca series (\ref{eq09}) can be written as
follows:
\begin{equation}
\gamma= \sum_{k=1}^{\infty}L_B(k)Q(k,B), \label{eq6}
\end{equation}
where $L_B(k):=\lfloor\log_BBk\rfloor
=\sum_{\alpha=0}^{B-1}N_{\alpha,B}(k)$ is the $B$-ary length of
$k.$ Indeed,  representing $n=Bk+r,$ $0\le r\le B-1$ and summing
in (\ref{eq09}) over $k\ge 1$ and $0\le r\le B-1$ we get
$$
\gamma=\sum_{k=1}^{\infty}\lfloor\log_BBk\rfloor\left(
\frac{B-1}{Bk}-\frac{1}{Bk+1}-\cdots-\frac{1}{Bk+B-1}\right)
=\sum_{k=1}^{\infty}\lfloor\log_BBk\rfloor Q(k,B).
$$
 Using the same notation, the generalized
Addison series (\ref{eq010}) gives another base-$B$ expansion of
Euler's constant
\begin{equation}
\gamma=\frac{1}{2}+\sum_{n=1}^{\infty}
\frac{L_B(n)P_B(n)}{Bn(Bn+1)\cdots (Bn+B)}
=\frac{1}{2}+\sum_{n=1}^{\infty}L_B(n)
\left(Q(n,B)-\frac{B-1}{2Bn(n+1)}\right) \label{eu}
\end{equation}
 which converges faster than (\ref{eq6}) to $\gamma.$ Here we used
the fact that
$$
\sum_{n=1}^{\infty}\sum_{\alpha=0}^{B-1}\frac{N_{\alpha,
B}(n)}{n(n+1)}=\frac{B}{B-1},
$$
which can be easily checked by \cite[Section 3]{as}. On the other
hand,
\begin{equation*}
\begin{split}
Q&(n,B)-\frac{B-1}{2Bn(n+1)}=\frac{1}{2}\sum_{m=1}^{B-1}\left(\frac{1}{Bn}-
\frac{2}{Bn+m}+\frac{1}{Bn+B}\right) \\
&=\frac{1}{Bn(Bn+B)}\sum_{m=1}^{B-1}
\left(2m-B+\frac{2m(B-m)}{Bn+m}\right)=\frac{P_B(n)}{Bn(Bn+1)\cdots(Bn+B)}.
\end{split}
\end{equation*}
Finally, we give a brief description of some other generalized Euler constants
that have appeared in the literature in Section \ref{S6}.



\section{Analytic continuation}
\label{S2}



We consider the generalized Euler constant function
$\gamma_{a,b}(z)$ defined in (\ref{eq7}), where $a, b$ are
positive real numbers, $z\in {\mathbb C},$ and the series
converges when $|z|\le 1.$ We show that $\gamma_{a,b}(z)$ admits
an analytic continuation to the domain ${\mathbb C}\setminus
[1,+\infty).$ The following theorem is a slight modification of
\cite[Theorem 3]{sh}.
\begin{theorem}
Let $a, b$ be positive real numbers, $z\in {\mathbb C},$ $|z|\le
1.$ Then
\begin{equation}
\gamma_{a,b}(z)=\int_0^1\int_0^1\frac{(xy)^{b-1}(1-x)}{(1-zx^ay^a)(-\log
xy)}\,dxdy=\int_0^1\frac{x^{b-1}(1-x)}{1-zx^a}\left(\frac{1}{1-x}+
\frac{1}{\log x}\right)\,dx. \label{eq8}
\end{equation}
The integrals converge for all $z\in {\mathbb C}\setminus
(1,+\infty)$ and give the analytic continuation of the
generalized Euler constant function $\gamma_{a,b}(z)$ for $z\in
{\mathbb C}\setminus [1,+\infty).$
\end{theorem}

\begin{proof} Denoting the double integral in (\ref{eq8}) by $I(z)$
and for $|z|\le 1,$ expanding $(1-zx^ay^a)^{-1}$ in a geometric
series we have
\begin{equation*}
\begin{split}
I(z)&=\sum_{k=0}^{\infty}z^k\int_0^1\int_0^1\frac{(xy)^{ak+b-1}(1-x)}{(-\log
xy)}\,dxdy \\
&=\sum_{k=0}^{\infty}z^k\int_0^1\int_0^1\int_0^{+\infty}(xy)^{t+ak+b-1}(1-x)\,dxdydt
\\
&=\sum_{k=0}^{\infty}z^k\int_0^{+\infty}\left(\frac{1}{(t+ak+b)^2}-\Bigl(
\frac{1}{t+ak+b}-\frac{1}{t+ak+b+1}\Bigr)\right)\,dt=\gamma_{a,b}(z).
\end{split}
\end{equation*}
On the other hand, making the change of variables $u=x^a,$ $v=y^a$
in the double integral we get
$$
I(z)=\frac{1}{a}\int_{0}^1\int_{0}^1\frac{(uv)^{\frac{b}{a}-1}(1-u^{\frac{1}{a}})}%
{(1-zuv)(-\log uv)}\,dudv.
$$
Now by \cite[Corollary 3.3]{gs}, for $z\in {\mathbb C}\setminus
[1,+\infty)$ we have
$$
I(z)=\frac{1}{a}\Phi\Bigl(z,1,\frac{b}{a}\Bigr)-\frac{\partial\Phi}{\partial
s}\Bigl(z,0,\frac{b}{a}\Bigr)+\frac{\partial\Phi}{\partial
s}\Bigl(z,0,\frac{b+1}{a}\Bigr),
$$
where $\Phi(z,s,u)$ is the Lerch transcendent, a holomorphic
function in $z$ and $s,$ for $z\in {\mathbb C}\setminus
[1,+\infty)$ and all complex $s$ (see \cite[Lemma 2.2]{gs}), which
is the analytic continuation of the series
$$
\Phi(z,s,u)=\sum_{n=0}^{\infty}\frac{z^n}{(n+u)^s}, \qquad u>0.
$$
To prove the second equality in (\ref{eq8}), make the change of
variables $X=xy,$ $Y=y$ and integrate with respect to $Y.$ 
\end{proof}

\begin{corollary} \label{c1}
Let $a,b$ be positive real numbers, $l\in {\mathbb N},$ $z\in
{\mathbb C}\setminus [1,+\infty).$ Then for the $l$-th derivative
we have
$$
\gamma_{a,b}^{(l)}(z)=\int_0^1\int_0^1\frac{(xy)^{al+b-1}(x-1)}%
{(1-zx^ay^a)^{l+1}\log xy}\,dxdy=
\int_0^1\frac{x^{la+b-1}(1-x)}{(1-zx^a)^{l+1}}
\left(\frac{1}{1-x}+\frac{1}{\log x}\right)\,dx.
$$
\end{corollary}

From Corollary \ref{c1}, \cite[Cor.3.3, 3.8, 3.9]{gs} and
\cite[Lemma 4]{assh} we get


\begin{corollary} \label{c1.1}
Let $a, b$ be positive real numbers, $z\in {\mathbb C}\setminus
[1, +\infty).$ Then the following equalities hold:

$$
\gamma_{a,b}(1)=\log\Gamma\Bigl(\frac{b+1}{a}\Bigr)-\log\Gamma\Bigl(
\frac{b}{a}\Bigr)-\frac{1}{a}\psi\Bigl(\frac{b}{a}\Bigr),
$$

$$
\gamma_{a,b}(z)=\frac{1}{a}\Phi\Bigl(z,1,\frac{b}{a}\Bigr)-
\frac{\partial\Phi}{\partial s}\Bigl(z,0,\frac{b}{a}\Bigr)
+\frac{\partial\Phi}{\partial s}\Bigl(z,0,\frac{b+1}{a}\Bigr),
$$

\begin{equation*}
\begin{split}
\gamma'_{a,b}(z)=&-\frac{b}{a^2}\Phi\Bigl(z,1,\frac{b}{a}+1\Bigr)+
\frac{1}{a(1-z)}+\frac{b}{a}\frac{\partial\Phi}{\partial
s}\Bigl(z,0,\frac{b}{a}+1\Bigr)-  \frac{\partial\Phi}{\partial
s}\Bigl(z,-1,\frac{b}{a}+1\Bigr)-
\\[10pt]
&\frac{b+1}{a}\frac{\partial\Phi}{\partial
s}\Bigl(z,0,\frac{b+1}{a}+1\Bigr)+\frac{\partial\Phi}{\partial s}
\Bigl(z,-1,\frac{b+1}{a}+1\Bigr),
\end{split}
\end{equation*}
where $\Phi(z,s,u)$ is the Lerch transcendent and
$\psi(x)=\frac{d}{dx}\log\Gamma(x)$ is the logarithmic derivative
of the gamma function.
\end{corollary}


\section{Catalan-type integrals for $\gamma_{a,b}^{(l)}(z).$}
\label{S3}


Berndt and Bowman \cite{bebo}  demonstrated that for $x>0$ and any integer
$B>1,$ 
\begin{equation}
\frac{1}{1-x}+\frac{1}{\log x}=\sum_{k=1}^{\infty}\frac{(B-1)+
(B-2)x^{\frac{1}{B^k}}+(B-3)x^{\frac{2}{B^k}}+\cdots+x^{\frac{B-2}{B^k}}}%
{B^k(1+x^{\frac{1}{B^k}}+x^{\frac{2}{B^k}}+\cdots+x^{\frac{B-1}{B^k}})}.
\label{eq9}
\end{equation}
The special cases $B=2,3$ of this equality can be found in
Ramanujan's third note book \cite[p.\ 364]{ra}. Using this key
formula we prove the following generalization of the integral
(\ref{eq08}).
\begin{theorem} \label{t2}
 Let $a, b, B$ be positive integers with $B>1,$ $l$ a non-negative integer.
 If either $z\in {\mathbb C}\setminus [1,+\infty)$ and $l\ge 1,$
 or $z\in {\mathbb C}\setminus (1,+\infty)$ and $l=0,$ then
 \begin{equation}
 \gamma_{a,b}^{(l)}(z)=\int_0^1\left(\frac{B}{1-x^B}-\frac{1}{1-x}\right)
 F_l(z,x)\,dx
 \label{eq10}
 \end{equation}
 where
 \begin{equation}
 F_l(z,x)=\sum_{k=1}^{\infty}\frac{x^{(b+al)B^k-1}(1-x^{B^k})}%
 {(1-zx^{aB^k})^{l+1}}.
 \label{eq017}
 \end{equation}
 \end{theorem}

\begin{proof}  First we note that the series of variable $x$ on the
right-hand side of (\ref{eq9})  converges uniformly on $[0,1],$
since the absolute value of its general term does not exceed
$\frac{B-1}{2B^{k-1}}.$ Then for $l\ge 0,$  multiplying both sides
of (\ref{eq9}) by $\frac{x^{la+b-1}(1-x)}{(1-zx^a)^{l+1}}$ and
integrating over $0\le x\le 1$ we get
$$
\gamma_{a,b}^{(l)}(z)=\sum_{k=1}^{\infty}\int_0^1\frac{x^{la+b-1}(1-x)}{(1-zx^a)^{l+1}}
\cdot\frac{(B-1)+(B-2)x^{\frac{1}{B^k}}+\cdots+x^{\frac{B-2}{B^k}}}%
{B^k(1+x^{\frac{1}{B^k}}+x^{\frac{2}{B^k}}+\cdots+x^{\frac{B-1}{B^k}})}\,dx.
$$
Replacing $x$ by $x^{B^k}$ in each integral we find
\begin{equation*}
\begin{split}
\gamma_{a,b}^{(l)}(z)&=\sum_{k=1}^{\infty}\int_0^1\frac{x^{(la+b)B^k-1}(1-x^{B^k})}%
{(1-zx^{aB^k})^{l+1}}\cdot\frac{(B-1)+(B-2)x+\cdots+x^{B-2}}{1+x+x^2+\cdots+x^{B-1}}\,dx
\\ &=\int_0^1\left(\frac{B}{1-x^B}-\frac{1}{1-x}\right)F_l(z,x)\,dx,
\end{split}
\end{equation*}
as required.
\end{proof}

From Theorem \ref{t2} we readily get a generalization of
Ramanujan's integral.
\begin{corollary} \label{c2}
Let $a, b, B$ be positive integers with $B>1,$ $l$ a non-negative integer.
 If either $z\in {\mathbb C}\setminus [1,+\infty)$ and $l\ge 1,$
 or $z\in {\mathbb C}\setminus (1,+\infty)$ and $l=0,$ then
\begin{equation}
\gamma_{a,b}^{(l)}(z)=\int_0^1\frac{x^{b+al-1}(1-x)}{(1-zx^a)^{l+1}}\,dx
+\int_0^1\left(\frac{Bx^{B}}{1-x^B}-\frac{x}{1-x}\right)F_l(z,x)\,dx.
\label{eq018}
\end{equation}
\end{corollary}

\begin{proof} First we note that the series (\ref{eq017}), considered as a sum of
 functions of the variable $x$  converges uniformly on $[0,
 1-\varepsilon]$ for any $\varepsilon>0.$  Then integrating
 termwise we have
 $$
 \int_0^{1-\varepsilon}F_l(z,x)\,dx=\sum_{k=1}^{\infty}
 \int_0^{1-\varepsilon}\frac{x^{(b+al)B^k-1}(1-x^{B^k})}%
 {(1-zx^{aB^k})^{l+1}}\,dx.
 $$
Making the change of variable $y=x^{B^k}$ in each integral we get
$$
\int_0^{1-\varepsilon}F_l(z,x)\,dx=\sum_{k=1}^{\infty}
\frac{1}{B^k} \int_0^{(1-\varepsilon)^{B^k}}\frac{y^{b+al-1}(1-y)}%
 {(1-zy^a)^{l+1}}\,dy.
 $$
Since the last series, considered as a series in the  variable $\varepsilon,$
converges uniformly on $[0,1],$ letting $\varepsilon$ tend to zero we get
\begin{equation}
\int_0^1F_l(z,x)\,dx=\frac{1}{B-1}\int_0^1
\frac{y^{b+al-1}(1-y)}{(1-zy^a)^{l+1}}\,dy. \label{eq019}
\end{equation}
Now from (\ref{eq10}) and (\ref{eq019}) it follows that
$$
\gamma_{a,b}^{(l)}(z)-\int_0^1
\frac{y^{b+al-1}(1-y)}{(1-zy^a)^{l+1}}\,dy= \int_0^1\left(
\frac{Bx^B}{1-x^B}-\frac{x}{1-x}\right)F_l(z,x)\,dx,
$$
and the proof is complete. 
\end{proof}
Averaging the formulas (\ref{eq10}) and  (\ref{eq018}), we get the
following generalization of the integral (\ref{eq065}).
\begin{corollary} \label{c3}
Let $a, b, B$ be positive integers with $B>1,$ $l$ a non-negative integer.
 If either $z\in {\mathbb C}\setminus [1,+\infty)$ and $l\ge 1,$
 or $z\in {\mathbb C}\setminus (1,+\infty)$ and $l=0,$ then
$$
\gamma_{a,b}^{(l)}(z)=\frac{1}{2}\int_0^1\frac{x^{b+al-1}(1-x)}{(1-zx^a)^{l+1}}\,dx
+\frac{1}{2}\int_0^1\left(\frac{B(1+x^B)}{1-x^B}-\frac{1+x}{1-x}\right)F_l(z,x)\,dx.
$$
\end{corollary}



\section{Vacca-type series for $\gamma_{a,b}(z)$ and $\gamma'_{a,b}(z).$}
\label{S4}


\begin{theorem} \label{t3}
Let $a,b,B$ be positive integers with $B>1,$  $z\in {\mathbb C},$ $|z|\le
1.$   Then for
 the generalized Euler constant function $\gamma_{a,b}(z)$, the
 following expansion is valid:
 $$
 \gamma_{a,b}(z)=\sum_{k=1}^{\infty}a_kQ(k,B)=
 \sum_{k=1}^{\infty}a_{\lfloor\frac{k}{B}\rfloor}\frac{\varepsilon(k)}{k},
 $$
 where $Q(k,B)$ is a rational function given by {\rm (\ref{eq5})},
  $\{a_k\}_{k=0}^{\infty}$ is a sequence defined by the
 generating function
 \begin{equation}
 G(z,x)=\frac{1}{1-x}\sum_{k=0}^{\infty}\frac{x^{bB^k}(1-x^{B^k})}%
 {1-zx^{aB^k}}=\sum_{k=0}^{\infty}a_kx^k
 \label{eq11}
 \end{equation}
 and $\varepsilon(k)$ is defined in {\rm (\ref{eq095})}.
 \end{theorem}
\begin{proof} For $l=0,$ rewrite (\ref{eq10}) in the form
$$
\gamma_{a,b}(z)=\int_0^1\frac{1-x^B}{x}\left(\frac{B}{1-x^B}-\frac{1}{1-x}\right)
G(z,x^B)\,dx
$$
where $G(z,x)$ is defined in (\ref{eq11}). Then, since $a_0=0,$
we have
\begin{equation}
\gamma_{a,b}(z)=\int_0^1(B-1-x-x^2-\cdots-x^{B-1})\sum_{k=1}^{\infty}a_kx^{Bk-1}\,
dx. \label{eq12}
\end{equation}
Expanding $G(z,x)$ in a power series of $x,$
$$
G(z,x)=\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}z^mx^{(am+b)B^k}(1+x+\cdots+x^{B^k-1}),
$$
we see that $a_k=O(\ln_B k).$ Therefore, by termwise integration
in (\ref{eq12}), which can be easily justified  by the same way as
in the proof of Corollary \ref{c2}, we get
\begin{equation*}
\begin{split}
\gamma_{a,b}(z)
&=\sum_{k=1}^{\infty}a_k\int_0^1[(x^{Bk-1}-x^{Bk})+(x^{Bk-1}-x^{Bk+1})+\cdots+
(x^{Bk-1}-x^{Bk+B-2})]\,dx \\
&=\sum_{k=1}^{\infty}a_kQ(k,B). 
\end{split}
\end{equation*}
\end{proof}
\begin{theorem}
Let $a,b,B$ be positive integers with $B>1,$  $z\in {\mathbb C},$ $|z|\le
1.$   Then for
 the generalized Euler constant function, the
 following expansion is valid:
 $$
 \gamma_{a,b}(z)=\int_0^1\frac{x^{b-1}(1-x)}{1-zx^a}\,dx-\sum_{k=1}^{\infty}a_k
 \widetilde{Q}(k,B),
 $$
 where
 \begin{equation*}
 \begin{split}
 \widetilde{Q}(k,B)&=\frac{B-1}{Bk(k+1)}-Q(k,B) \\
&= \frac{B-1}{(Bk+B)(Bk+1)}+\frac{B-2}{(Bk+B)(Bk+2)}+\cdots+
 \frac{1}{(Bk+B)(Bk+B-1)}
 \end{split}
 \end{equation*}
  and the sequence
 $\{a_k\}_{k=1}^{\infty}$ is  defined in Theorem {\rm\ref{t3}}.
\end{theorem}

\begin{proof} From Corollary \ref{c2} with $l=0,$ using the same method
as in the proof of Theorem \ref{t3},
 we get
\begin{equation*}
\begin{split}
&\int_0^1\left(\frac{Bx^B}{1-x^B}-\frac{x}{1-x}\right)F_0(z,x)=
\int_0^1\frac{1-x^B}{x}\left(\frac{Bx^B}{1-x^B}-\frac{x}{1-x}\right)G(z,x^B)\,dx
\\
&=\int_0^1(Bx^{B-1}-(1+x+\cdots+x^{B-1}))\sum_{k=1}^{\infty}a_kx^{Bk}\,dx
\\ &=\sum_{k=1}^{\infty}a_k\int_0^1[(x^{Bk+B-1}-x^{Bk+B-2})+\cdots+
(x^{Bk+B-1}-x^{Bk+1})+(x^{Bk+B-1}-x^{Bk})]\,dx \\
&=-\sum_{k=1}^{\infty}a_k\widetilde{Q}(k,B). 
\end{split}
\end{equation*}
\end{proof}
\begin{theorem} \label{t5}
Let $a,b,B$ be positive integers with $B>1,$  $z\in {\mathbb C},$ $|z|\le
1.$   Then for
 the generalized Euler constant function $\gamma_{a,b}(z)$ and its derivative, the
 following expansion is valid:
$$
\gamma_{a,b}^{(l)}(z)=\frac{1}{2}\int_0^1\frac{x^{b+al-1}(1-x)}{(1-zx^a)^{l+1}}\,dx
+\sum_{k=1}^{\infty}a_{k,l}\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)},
\qquad l=0,1,
$$
where $P_B(k)$ is a polynomial of degree $B-2$ given by {\rm
(\ref{polyn})}, $z\ne 1$ if $l=1,$ and   the sequence
$\{a_{k,l}\}_{k=0}^{\infty}$ is defined by the generating function
\begin{equation}
G_l(z,x)=\frac{1}{1-x}\sum_{k=0}^{\infty}\frac{x^{(b+al)B^k}(1-x^{B^k})}%
{(1-zx^{aB^k})^{l+1}}=\sum_{k=0}^{\infty}a_{k,l}x^k, \qquad l=0,
1. \label{eq24}
\end{equation}
\end{theorem}

\begin{proof} Expanding $G_l(z,x)$ in a power series of $x,$
$$
G_l(z,x)=\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\binom{m+l}{l}z^mx^{(b+al+am)B^k}(1+x+x^2+
\cdots+x^{B^k-1}),
$$
we see that $a_{k,l}=O(k^l\ln_Bk).$ Therefore, for $l=0,1,$ by
termwise integration we get
\begin{equation*}
\begin{split}
&\int_0^1\left(\frac{B(1+x^B)}{1-x^B}-\frac{1+x}{1-x}\right)F_l(z,x)dx
=\int_0^1\frac{1-x^B}{x}\left(\frac{B(1+x^B)}{1-x^B}-\frac{1+x}{1-x}\right)G_l(z,x^B)dx
\\
&=\int_0^1[(B-1)-2x-2x^2-\cdots-2x^{B-1}+(B-1)x^B]\sum_{k=1}^{\infty}a_{k,l}x^{Bk-1}\,dx
\\
&=\sum_{k=1}^{\infty}a_{k,l}\left(\frac{B-1}{Bk}-\frac{2}{Bk+1}-\frac{2}{Bk+2}-\cdots
-\frac{2}{Bk+B-1}+\frac{B-1}{Bk+B}\right) \\
&=2\sum_{k=1}^{\infty}a_{k,l}\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)},
\end{split}
\end{equation*}
where $P_B(k)$ is defined in (\ref{polyn}) and the last series
converges  since $\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)}=O(k^{-3}).$
Now our theorem easily follows from Corollary \ref{c3}. 
\end{proof}





\section{Examples of rational series}
\label{S5}


It is easily seen that the generating function (\ref{eq24})
satisfies the following  functional equation:
\begin{equation}
G_l(z,x)-\frac{1-x^B}{1-x}G_l(z,x^B)=\frac{x^{b+al}}{(1-zx^a)^{l+1}},
 \label{eq25}
\end{equation}
which is equivalent to the following identity for series:
$$
\sum_{k=0}^{\infty}a_{k,l}x^k-(1+x+\cdots+x^{B-1})\sum_{k=0}^{\infty}a_{k,l}x^{Bk}
=\sum_{k=l}^{\infty}\binom{k}{l}z^{k-l}x^{ak+b}.
$$
Comparing coefficients of powers of $x$ we get an alternative
definition of the sequence $\{a_{k,l}\}_{k=0}^{\infty}$ by means
of the recursion
$$
a_{0,l}=a_{1,l}=\cdots=a_{al+b-1,l}=0
$$
and for $k\ge al+b,$
\begin{equation}
a_{k,l}=\begin{cases} a_{\lfloor\frac{k}{B}\rfloor,l}, &
\qquad\text{if} \qquad
k\not\equiv b \pmod a; \\
a_{\lfloor\frac{k}{B}\rfloor,l}+\binom{(k-b)/a}{l}z^{\frac{k-b}{a}-l},
& \qquad\text{if} \qquad  k\equiv b\pmod a.
\end{cases}
\label{re}
\end{equation}
On the other hand, in view of Corollary \ref{c1.1},
$\gamma_{a,b}(z)$ and $\gamma'_{a,b}(z)$ can be explicitly
expressed in terms of the Lerch transcendent, $\psi$-function and
logarithm of the gamma function.  This allows us to sum the series
 in Theorems \ref{t3}--\ref{t5} in terms of these functions.


\begin{example}
Suppose that $\omega$ is a non-empty word over
the alphabet $\{0,1,\ldots,B-1\}.$ Then obviously $\omega$ is
uniquely defined by its length $|\omega|$ and its size
$v_B(\omega)$ which is the value of $\omega$ when interpreted as
an integer in base $B.$ Let $N_{\omega,B}(k)$ be the number of
(possibly overlapping) occurrences of the block $\omega$ in the
$B$-ary expansion of $k.$ Note that for every $B$ and $\omega,$
$N_{\omega,B}(0)=0,$ since the $B$-ary expansion of zero is the
empty word. If the word $\omega$ begins with $0,$ but
$v_B(\omega)\ne 0,$ then in computing $N_{\omega,B}(k)$ we assume
that the $B$-ary expansion of $k$ starts with an arbitrary long
prefix of $0$'s. If $v_B(\omega)=0$ we take for $k$ the usual
shortest $B$-ary expansion of $k.$

Now we consider equation (\ref{eq25}) with  $l=0,$ $z=1$
\begin{equation}
G(1,x)-\frac{1-x^B}{1-x}G(1,x^B)=\frac{x^b}{1-x^a} \label{eq26}
\end{equation}
  and for a given non-empty word $\omega,$ set  $a=B^{|\omega|}$  in
(\ref{eq26}) and
\begin{equation*}
b=\begin{cases}
B^{|\omega|},     & \quad \text{if}\quad v_B(\omega)=0; \\
v_B(\omega), & \quad \text{if}\quad v_B(\omega)\ne 0.
 \end{cases}
\end{equation*}
Then by (\ref{re}), it is easily seen that
$a_k:=a_{k,0}=N_{\omega, B}(k),$ $k=1,2,\ldots,$ and by Theorem
\ref{t3}, we get another  proof of the following statement (see
\cite[Sections 3, 4.2]{assh}).
\end{example}

\begin{corollary} \label{c4}
Let $\omega$ be a non-empty word over the alphabet $\{0,1,\ldots,
B-1\}.$ Then
\begin{equation*}
\sum_{k=1}^{\infty}N_{\omega, B}(k)Q(k,B)=\begin{cases}
\gamma_{B^{|\omega|},v_B(\omega)}(1),     & \quad \text{if}\quad v_B(\omega)\ne 0; \\
\gamma_{B^{|\omega|}, B^{|\omega}|}(1), & \quad \text{if}\quad
v_B(\omega)= 0.
 \end{cases}
\end{equation*}
\end{corollary}
By Corollary \ref{c1.1}, the right-hand side of the last equality
can be calculated explicitly and we have
\begin{equation}
\sum_{k=1}^{\infty}N_{\omega, B}(k)Q(k,B)=\begin{cases}
\log\Gamma\left(\frac{v_B(\omega)+1}{B^{|\omega|}}\right)-
\log\Gamma\left(\frac{v_B(\omega)}{B^{|\omega|}}\right)-
\frac{1}{B^{|\omega|}}\psi\left(\frac{v_B(\omega)}{B^{|\omega|}}\right),     &  \text{if}\,\, v_B(\omega)\ne 0; \\
\log\Gamma\left(\frac{1}{B^{|\omega|}}\right)+
\frac{\gamma}{B^{|\omega|}}-|\omega|\log B, &  \text{if}\,\,
v_B(\omega)= 0.
 \end{cases}
 \label{eq28}
\end{equation}
\begin{corollary} \label{c54}
Let $\omega$ be a non-empty word over the alphabet $\{0,1,\ldots,
B-1\}.$ Then
\begin{equation*}
\begin{split}
&\sum_{k=1}^{\infty}\frac{N_{\omega, B}(k)P_B(k)}{Bk(Bk+1)\cdots
(Bk+B)} \\ &=\begin{cases}
\gamma_{B^{|\omega|},v_B(\omega)}(1)-\frac{1}{2B^{|\omega|}}
\left(\psi\Bigl(\frac{v_B(\omega)+1}{B^{|\omega|}}\Bigr)
-\psi\Bigl(\frac{v_B(\omega)}{B^{|\omega|}}\Bigr)\right),     & \quad \text{if}\quad v_B(\omega)\ne 0; \\
\gamma_{B^{|\omega|}, B^{|\omega}|}(1)-
\frac{1}{2B^{|\omega|}}\psi\Bigl(\frac{1}{B^{|\omega|}}\Bigr)-
\frac{\gamma}{2B^{|\omega|}}-\frac{1}{2}, & \quad \text{if}\quad
v_B(\omega)= 0.
 \end{cases}
 \end{split}
\end{equation*}
\end{corollary}
\begin{proof} The required statement easily follows from Theorem
\ref{t5}, Corollary \ref{c4} and the equality
$$
\int_0^1\frac{x^{b-1}(1-x)}{1-x^a}\,dx=\sum_{k=0}^{\infty}
\left(\frac{1}{ak+b}-\frac{1}{ak+b+1}\right)=\frac{1}{a}
\left(\psi\Bigl(\frac{b+1}{a}\Bigr)-\psi\Bigl(\frac{b}{a}\Bigr)\right).
$$
\end{proof}
From Theorem \ref{t3}, (\ref{eq25}) and (\ref{re}) with $a=1,$
$l=0$ we have
\begin{corollary} \label{c5}
Let $b, B$ be positive integers with $B>1,$  $z\in {\mathbb C},$ $|z|\le 1.$
Then
$$
\gamma_{1,b}(z)=\sum_{k=1}^{\infty}a_kQ(k,B)=\sum_{k=1}^{\infty}a_{\lfloor\frac{k}{B}\rfloor}
\frac{\varepsilon(k)}{k},
$$
where $a_0=a_1=\cdots=a_{b-1}=0,$
$a_k=a_{\lfloor\frac{k}{B}\rfloor}+z^{k-b},$ $k\ge b.$
\end{corollary}

Similarly, from Theorem \ref{t5} we have
\begin{corollary} \label{c65}
Let $b,B$ be positive integers with $B>1,$ $z\in {\mathbb C},$ $|z|\le 1.$
Then
$$
\gamma_{1,b}(z)=\frac{1}{2}\sum_{k=0}^{\infty}\frac{z^k}{(k+b)(k+b+1)}
+\sum_{k=1}^{\infty}a_k\frac{P_B(k)}{Bk(Bk+1)\cdots (Bk+B)},
$$
where $a_0=a_1=\cdots=a_{b-1}=0,$
$a_k=a_{\lfloor\frac{k}{B}\rfloor}+z^{k-b},$ $k\ge b.$
\end{corollary}

\begin{example}
If in Corollary \ref{c5} we take $z=1,$ then we
get that  $a_k$ is equal to the   $B$-ary length of
$\lfloor\frac{k}{b}\rfloor,$ i. e.,
$$
a_k=\sum_{\alpha=0}^{B-1}N_{\alpha,B}\left(\Bigl\lfloor\frac{k}{b}\Bigr\rfloor\right)
=L_B\left(\Bigl\lfloor\frac{k}{b}\Bigr\rfloor\right).
$$
On the other hand,
$$
\gamma_{1,b}(1)=\log b-\psi(b)=\log
b-\sum_{k=1}^{b-1}\frac{1}{k}+\gamma
$$
and hence we get
\begin{equation}
\log
b-\psi(b)=\sum_{k=1}^{\infty}L_B\left(\Bigl\lfloor\frac{k}{b}\Bigr\rfloor
\right)Q(k,B). \label{eq29}
\end{equation}
If $b=1,$ formula (\ref{eq29}) gives (\ref{eq6}). If $b>1,$ then
from (\ref{eq29}) and (\ref{eq6}) we get
\begin{equation}
\log
b=\sum_{k=1}^{b-1}\frac{1}{k}+\sum_{k=1}^{\infty}\left(L_B\left(\Bigl\lfloor
\frac{k}{b}\Bigr\rfloor\right)-L_B(k)\right)Q(k,B), \label{eq30}
\end{equation}
which is equivalent to \cite[Theorem 2.8]{bebo}. Similarly, from
Corollary \ref{c65} we obtain (\ref{eu}) and
\begin{equation}
\log
b=\sum_{k=1}^{b-1}\frac{1}{k}-\frac{b-1}{2b}+\sum_{k=1}^{\infty}
\frac{\left(L_B(\lfloor
\frac{k}{b}\rfloor)-L_B(k)\right)P_B(k)}%
{Bk(Bk+1)\cdots (Bk+B)}. \label{eq30.5}
\end{equation}
\end{example}

\begin{example}
Using the fact that for any integer $B>1,$
$$
L_B\left(\Bigl\lfloor\frac{k}{B}\Bigr\rfloor\right)-L_B(k)=-1,
$$
from (\ref{eq28}), (\ref{eq6}) and (\ref{eq30}) we get the
following rational series for $\log\Gamma(1/B):$
$$
\log\Gamma\left(\frac{1}{B}\right)=\sum_{k=1}^{B-1}\frac{1}{k}
+\sum_{k=1}^{\infty}\Bigl(N_{0,B}(k)-\frac{1}{B}L_B(k)-1\Bigr)Q(k,B).
$$
\end{example}

\begin{example}
Substituting $b=1,$ $z=-1$ in Corollary \ref{c5}
we get the generalized Vacca series for $\log\frac{4}{\pi}.$
\end{example}
\begin{corollary} \label{c6}
 Let $B\in {\mathbb N},$ $B>1.$  Then
$$
\log\frac{4}{\pi}=\sum_{k=1}^{\infty}a_kQ(k,B)=\sum_{k=1}^{\infty}a_{\lfloor\frac{k}{B}\rfloor}
\frac{\varepsilon(k)}{k},
$$
where
\begin{equation}
a_0=0, \qquad a_k=a_{\lfloor\frac{k}{B}\rfloor}+(-1)^{k-1}, \quad
k\ge 1. \label{eq31}
\end{equation}
In particular, if $B$ is even, then
\begin{equation}
\log\frac{4}{\pi}=\sum_{k=1}^{\infty}(N_{odd,B}(k)-N_{even,B}(k))Q(k,B)
=\sum_{k=1}^{\infty}
\frac{\left(N_{odd,B}(\lfloor\frac{k}{B}\rfloor)-
N_{even,B}(\lfloor\frac{k}{B}\rfloor)\right)}{k} \varepsilon(k),
\label{eq32}
\end{equation}
where $N_{odd,B}(k)$ (respectively $N_{even,B}(k)$) is the number
of occurrences  of the odd (respectively even) digits in the
$B$-ary expansion of $k.$
\end{corollary}
\begin{proof} To prove (\ref{eq32}), we notice that if $B$ is even,
then the sequence $\widetilde{a}_k:=N_{odd,B}(k)-N_{even,B}(k)$
satisfies  recurrence (\ref{eq31}).
\end{proof}

Substituting $b=1, z=-1$ in Corollary \ref{c65} with the help of
(\ref{eq30.5}) we get the generalized Addison series for
$\log\frac{4}{\pi}.$
\begin{corollary} \label{c69}
Let $B>1$ be a positive integer. Then
$$
\log\frac{4}{\pi}=\frac{1}{4}+\sum_{k=1}^{\infty}\frac{\left(L_B(\lfloor
\frac{k}{2}\rfloor)-L_B(k)+a_k\right)P_B(k)}{Bk(Bk+1)\cdots
(Bk+B)},
$$
where the sequence $a_k$ is defined in Corollary {\rm\ref{c6}}. In
particular, if $B$ is even, then
$$
\log\frac{4}{\pi}=\frac{1}{4}+\sum_{k=1}^{\infty}\frac{\left(L_B(\lfloor
\frac{k}{2}\rfloor)-2N_{even,B}(k)\right)P_B(k)}{Bk(Bk+1)\cdots
(Bk+B)}.
$$
\end{corollary}
\begin{example}
For $t>1,$ the generalized Somos constant
$\sigma_t$ is defined by
$$
\sigma_t=\sqrt[t]{1\sqrt[t]{2\sqrt[t]{3\cdots}}}=
1^{1/t}2^{1/t^2}3^{1/t^3}\cdots=\prod_{n=1}^{\infty}n^{1/t^n}
$$
(see \cite[Section 3]{sh}).
In view of the relation \cite[Theorem 8]{sh}
\begin{equation}
\gamma_{1,1}\left(\frac{1}{t}\right)=t\log\frac{t}%
{(t-1)\sigma_t^{t-1}}, \label{somos}
\end{equation}
by Corollary  \ref{c5} and formula (\ref{eq30}) we get
\end{example}
\begin{corollary} \label{c7}
Let $B\in {\mathbb N},$ $B>1,$ $t\in {\mathbb R},$ $t>1.$ Then
$$
\log\sigma_t=\frac{1}{(t-1)^2}+\frac{1}{t-1}\sum_{k=1}^{\infty}
\left(L_B\Bigl(\Bigl\lfloor\frac{k}{t}\Bigr\rfloor\Bigr)-L_B\Bigl(\Bigl\lfloor
\frac{k}{t-1}\Bigr\rfloor\Bigr)-\frac{a_k}{t}\right)Q(k,B),
$$
where $a_0=0,$ $a_k=a_{\lfloor\frac{k}{B}\rfloor}+t^{1-k},$ $k\ge
1.$
\end{corollary}
In particular, setting $B=t=2$ we get the following rational
series for Somos's quadratic recurrence constant:
$$
\log\sigma_2=1-\frac{1}{2}\sum_{k=1}^{\infty}\frac{b_k}{2k(2k+1)},
$$
where $b_1=3,$
$b_k=b_{\lfloor\frac{k}{2}\rfloor}+\frac{1}{2^{k-1}},$ $k\ge 2.$

From (\ref{somos}), (\ref{eq30.5}) and Theorem \ref{t5} we find
\begin{corollary} \label{c8}
Let $B\in {\mathbb N},$ $B>1,$ $t\in {\mathbb R},$ $t>1.$ Then
\begin{equation*}
\begin{split}
&\log\sigma_t=\frac{3t-1}{4t(t-1)^2} \\
&+\frac{t+1}{2(t-1)}\sum_{k=1}^{\infty}
\left(L_B\Bigl(\Bigl\lfloor\frac{k}{t}\Bigr\rfloor\Bigr)-L_B\Bigl(\Bigl\lfloor
\frac{k}{t-1}\Bigr\rfloor\Bigr)-\frac{2a_k}{t(t+1)}\right)
\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)},
\end{split}
\end{equation*}
 where the sequence $a_k$ is
defined in Corollary {\rm \ref{c7}}.
\end{corollary}
In particular, if $B=t=2$ we get
$$
\log\sigma_2=\frac{5}{8}-\frac{1}{2}\sum_{k=1}^{\infty}
\frac{c_k}{2k(2k+1)(2k+2)},
$$
where $c_1=4,$
$c_k=c_{\lfloor\frac{k}{2}\rfloor}+\frac{1}{2^{k-1}},$ $k\ge 2.$

\begin{example}
The Glaisher-Kinkelin constant is defined by the
limit \cite[p.135]{fi}
$$
A:=\lim_{n\to\infty}\frac{1^22^2\cdots
n^n}{n^{\frac{n^2+n}{2}+\frac{1}{12}}e^{-\frac{n^2}{4}}}=1.28242712\cdots.
$$
Its connection to the generalized Euler constant function
$\gamma_{a,b}(z)$ is given by the formula \cite[Corollary 4]{sh}
\begin{equation}
\gamma'_{1,1}(-1)=\log\frac{2^{11/6}A^6}{\pi^{3/2}e}.
\label{glaisher}
\end{equation}
By Theorem \ref{t5}, since
$$
\int_0^1\frac{x(1-x)}{(1+x)^2}\,dx=3\log 2-2,
$$
we have
$$
\log A=\frac{4}{9}\log 2-\frac{1}{4}\log\frac{4}{\pi}+
\frac{1}{6}\sum_{k=1}^{\infty}a_{k,1}\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)},
$$
where the sequence $a_{k,1}$ is defined by the generating function
(\ref{eq24}) with $a=b=l=1,$ $z=-1,$ or using   the
recursion (\ref{re}):
$$
a_{0,1}=a_{1,1}=0, \qquad
a_{k,1}=a_{\lfloor\frac{k}{B}\rfloor,1}+(-1)^k(k-1), \quad k\ge 2.
$$
Now by Corollary \ref{c69} and (\ref{eq30.5}) we get
\end{example}
\begin{corollary} \label{c12}
Let $B>1$ be a positive integer. Then
$$
\log A=\frac{13}{48}-\frac{1}{36}\sum_{k=1}^{\infty}
\left(7L_B(k)-7L_B\Bigl(\Bigl\lfloor\frac{k}{2}\Bigr\rfloor\Bigr)
+b_k\right)\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)},
$$
where $b_0=0,$
$b_k=b_{\lfloor\frac{k}{B}\rfloor}+(-1)^{k-1}(6k+3),$ $k\ge 1.$
\end{corollary}
In particular, if $B=2$ we get
$$
\log A=\frac{13}{48}-\frac{1}{36}\sum_{k=1}^{\infty}
\frac{c_k}{2k(2k+1)(2k+2)},
$$
where $c_1=16,$
$c_k=c_{\lfloor\frac{k}{2}\rfloor}+(-1)^{k-1}(6k+3),$ $k\ge 2.$

Using the formula expressing $\frac{\zeta'(2)}{\pi^2}$ in terms of the
Glaisher-Kinkelin constant \cite[p.\ 135]{fi},
$$
\log A=-\frac{\zeta'(2)}{\pi^2}+\frac{\log 2\pi+\gamma}{12},
$$
by Corollaries \ref{c65}, \ref{c69} and \ref{c12}, we get
\begin{corollary}
Let $B>1$ be a positive integer. Then
$$
\frac{\zeta'(2)}{\pi^2}=-\frac{1}{16}+\frac{1}{36}\sum_{k=1}^{\infty}
\left(4L_B(k)-L_B\Bigl(\Bigl\lfloor\frac{k}{2}\Bigr\rfloor\Bigr)+c_k\right)
\frac{P_B(k)}{Bk(Bk+1)\cdots(Bk+B)},
$$
where $c_0=0,$ $c_k=c_{\lfloor\frac{k}{B}\rfloor}+(-1)^{k-1}6k,$
$k\ge 1.$
\end{corollary}

\begin{example}
First we evaluate $\gamma_{2,1}^{(l)}(-1)$ for
$l=0, 1.$ From Corollaries  \ref{c1}, \ref{c1.1} and
\cite[Examples 3.9, 3.15]{gs} we have
$$
\gamma_{2,1}(-1)=\int_0^1\int_0^1\frac{(x-1)\,dxdy}{(1+x^2y^2)\log
xy}=\frac{\pi}{4}-2\log\Gamma\Bigl(\frac{1}{4}\Bigr)+\log\sqrt{2\pi^3}
$$
and
\begin{equation*}
\begin{split}
\gamma'_{2,1}(-1)=&-\frac{1}{4}\Phi(-1,1,3/2)+\frac{1}{2}
\Phi(-1,0,3/2) +\frac{1}{2}\frac{\partial\Phi}{\partial
s}(-1,0,3/2) \\
&-\frac{\partial\Phi}{\partial s}(-1,-1,3/2)-
\frac{\partial\Phi}{\partial
s}(-1,0,2)+\frac{\partial\Phi}{\partial s}(-1,-1,2).
\end{split}
\end{equation*}
The last expression
can be evaluated explicitly (see \cite[Section 2]{gs}) and we get
$$
\gamma'_{2,1}(-1)=\frac{G}{\pi}+\frac{\pi}{8}-\log\Gamma\Bigl(
\frac{1}{4}\Bigr)-3\log A+\log\pi+\frac{1}{3}\log 2,
$$
or
\begin{equation}
\frac{G}{\pi}=\gamma'_{2,1}(-1)-\frac{1}{2}\gamma_{2,1}(-1)
+\frac{1}{4}\log\frac{4}{\pi}+3\log A-\frac{7}{12}\log 2.
\label{eq101}
\end{equation}
On the other hand, by Theorem \ref{t5} and (\ref{re}) we have
\begin{equation}
\gamma_{2,1}(-1)=\frac{\pi}{8}-\frac{1}{4}\log
2+\sum_{k=1}^{\infty}a_{k,0}\frac{P_B(k)}{Bk(Bk+1)\cdots (Bk+B)},
\label{eq102}
\end{equation}
where $a_{0,0}=0,$ $a_{2k,0}=a_{\lfloor\frac{2k}{B}\rfloor,0},$
$k\ge 1,$ $a_{2k+1,0}=a_{\lfloor\frac{2k+1}{B}\rfloor,0}+(-1)^k,$
$k\ge 0,$ and
\begin{equation}
\gamma'_{2,1}(-1)=\frac{\pi}{16}-\frac{1}{4}\log
2+\sum_{k=1}^{\infty}a_{k,1}\frac{P_B(k)}{Bk(Bk+1)\cdots (Bk+B)},
\label{eq103}
\end{equation}
where $a_{0,1}=0,$ $a_{2k,1}=a_{\lfloor\frac{2k}{B}\rfloor,1},$
$k\ge 1,$
$a_{2k+1,1}=a_{\lfloor\frac{2k+1}{B}\rfloor,1}+(-1)^{k-1}k,$ $k\ge
0.$ Now from (\ref{eq101}) -- (\ref{eq103}), (\ref{eq30.5}) and
Corollary \ref{c69} we get the following expansion for $G/\pi.$
\end{example}
\begin{corollary} Let $B>1$ be a positive integer. Then
$$
\frac{G}{\pi}=\frac{11}{32}+\sum_{k=1}^{\infty}\left(
\frac{1}{8}L_B\Bigl(\Bigl\lfloor\frac{k}{2}\Bigr\rfloor\Bigr)
-\frac{1}{8}L_B(k)+c_k\right)\frac{P_B(k)}{Bk(Bk+1)\cdots (Bk+B)},
$$
where $c_0=0,$ $c_{2k}=c_{\lfloor\frac{2k}{B}\rfloor}+k,$ $k\ge
1,$ $c_{2k+1}=c_{\lfloor\frac{2k+1}{B}\rfloor}+
\frac{(-1)^{k-1}-1}{2}(2k+1),$ $k\ge 0.$
\end{corollary}
In particular, if $B=2$ we get
$$
\frac{G}{\pi}=\frac{11}{32}+\sum_{k=1}^{\infty}\frac{b_k}{2k(2k+1)(2k+2)},
$$
where $b_1=-\frac{9}{8},$ $b_{2k}=b_k+k,$ $b_{2k+1}=b_k+
\frac{(-1)^{k-1}-1}{2}(2k+1),$ $k\ge 1.$


\section{Other generalized Euler constants}
\label{S6}


The purpose of this section is to draw attention to different generalizations
of Euler's constant for which many interesting results remain to be discovered.

The simplest way to generalize Euler's constant
\begin{equation}
\gamma=\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k}-\log n\right),
\label{6.1}
\end{equation}
which is related to the digamma function by the equality $\gamma=-\psi(1),$
is to consider for $0<\alpha\le 1,$
$$
\gamma(\alpha)=\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k+\alpha}-\log n\right)=
\lim_{n\to\infty}\left(\sum_{k=1}^n\frac{1}{k+\alpha}-\log(n+\alpha)\right).
$$
Tasaka \cite{ta}  proved that $\gamma(\alpha)=-\psi(\alpha).$
Its connection to the generalized Euler constant function $\gamma_{a,b}(z)$
is given by the formula
$$
\gamma(\alpha)+\log\alpha=\gamma_{1,\alpha}(1).
$$
Briggs \cite{br} and Lehmer \cite{l}  studied the analog of $\gamma$
corresponding to the arithmetical progression of positive integers
$r,$ $r+m,$ $r+2m, \ldots,$ $(r\le m):$
$$
\gamma(r,m)=\lim_{n\to\infty}\left(H(n,r,m)-\frac{1}{m}\log n\right),
$$
where $H(n,r,m)=\underset{k\equiv r\!\!\!\pmod m}{\sum\limits_{0<k\le n, }}\frac{1}{k}.$
Since $H(n,r,m)=\sum\limits_{0\le k\le (n-r)/m}\frac{1}{r+mk},$ it is easily seen that
$$
m\gamma(r,m)=\gamma(r/m)-\log m=\gamma_{1,r/m}(1)-\log r.
$$
Diamond and Ford \cite{df} studied a family $\{\gamma({\mathcal{P}})\}$ of generalized
Euler constants arising from the sum of reciprocals of integers sieved by finite sets of primes
${\mathcal{P}}.$ More precisely, if ${\mathcal{P}}$ represents a finite set of primes, let
$$
1_{\mathcal{P}}(n):=\begin{cases}
1,     &  \text{if}\quad  \gcd\left(n, \prod_{p\in{\mathcal{P}}}p\right)=1; \\
 0, &  \text{otherwise};
 \end{cases}
 \quad\text{and}\quad \delta_{\mathcal{P}}:=\lim_{x\to\infty}\frac{1}{x}
 \sum_{n\le x}1_{\mathcal{P}}(n).
$$
A simple argument shows that $\delta_{\mathcal{P}}=\prod_{p\in{\mathcal{P}}}(1-1/p)$
and that the generalized Euler constant
$$
\gamma({\mathcal{P}}):=\lim_{x\to\infty}\left(\sum_{n\le x}
\frac{1_{\mathcal{P}}(n)}{n}-\delta_{\mathcal{P}}\log x\right)
$$
exists. Its connection to Euler's constant is given by the formula \cite{df}
$$
\gamma({\mathcal{P}})=\prod_{p\in{\mathcal{P}}}\left(1-\frac{1}{p}\right)\left\{
\gamma+\sum_{p\in{\mathcal{P}}}\frac{\log p}{p-1}\right\}.
$$
Another generalization of the Euler constant is connected with the well-known limit
involving the Riemann zeta function:
\begin{equation}
\gamma=\lim_{s\to 1}\left(\zeta(s)-\frac{1}{s-1}\right).
\label{6.2}
\end{equation}
Expanding the Riemann zeta function into Laurent series in a neighborhood of its simple
pole at $s=1$ gives
\begin{equation*}
\zeta(s)=\frac{1}{s-1}+\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}\gamma_k(s-1)^k.
\end{equation*}
Stieltjes \cite{st} pointed out that the coefficients $\gamma_k$ can be expressed as
\begin{equation}
\gamma_k=\lim_{n\to\infty}\left(\sum_{j=1}^n\frac{\log^k j}{j}-\frac{\log^{k+1} n}{k+1}\right),
\qquad k=0,1,2,\ldots.
\label{6.3}
\end{equation}
(In the case $k=0,$ the first summand requires evaluation of $0^0,$ which is taken to be $1.$)
The coefficients $\gamma_k$ are usually called Stieltjes, or generalized Euler, constants (see
\cite{dil, fi, kre}.
In particular, the zero'th constant $\gamma_0=\gamma$ is the Euler constant.

Hardy \cite{ha} gave an analog of the Vacca series (\ref{eq3}) for $\gamma_1$ containing logarithmic coefficients:
$$
\gamma_1=\sum_{k=1}^{\infty}(-1)^k\frac{\log (k)\lfloor\log_2(k)\rfloor}{k}
-\frac{\log 2}{2}\sum_{k=1}^{\infty}(-1)^k\frac{\lfloor\log_2(2k)\rfloor\lfloor\log_2(k)\rfloor}{k},
$$
and Kluyver \cite{kl} presented more such series for higher-order constants.

The analog of $\gamma_k$ corresponding to the arithmetical progression
$r,$ $r+m,$ $r+2m, \ldots$ was studied by Knopfmacher \cite{kn}, Kanemitsu \cite{ka}, and Dilcher \cite{dil}:
$$
\gamma_k(r,m)=\lim_{n\to\infty}\Biggl(\underset{j\equiv r\!\!\!\!\!\pmod m}{\sum_{0<j\le n}}
\frac{\log^k j}{j}-\frac{1}{m}\frac{\log^{k+1} n}{k+1}\Biggr).
$$
Another extension of $\gamma_k$ can be derived from the Laurent series expansion of the Hurwitz
zeta function:
$$
\zeta(s,\alpha):=\sum_{n=0}^{\infty}\frac{1}{(n+\alpha)^s}=\frac{1}{s-1}+
\sum_{k=0}^{\infty}\frac{(-1)^k\gamma_k(\alpha)}{k!}(s-1)^k.
$$
Here $0<\alpha\le 1.$ Since $\zeta(s,1)=\zeta(s),$ we have $\gamma_k(1)=\gamma_k.$ Berndt \cite{be}
showed that
$$
\gamma_k(\alpha)=\lim_{n\to\infty}\left(\sum_{j=0}^n\frac{\log^k(j+\alpha)}{(j+\alpha)}-
\frac{\log^{k+1}(n+\alpha)}{k+1}\right),
$$
which is equivalent to (\ref{6.3}) when $\alpha=1.$ If $k=0$ and $\alpha=r/m,$ $r,m \in{\mathbb N},$
$r\le m,$ then
$$
\gamma_0(r/m)=m\gamma(r,m)+\log m=\gamma_{1,r/m}(1)-\log(r/m)=\gamma(r/m)=-\psi(r/m).
$$
Recently, Lampret \cite{la} considered the zeta-generalized Euler constant function
\begin{equation}
\Upsilon(s):=\sum_{j=1}^{\infty}\left(\frac{1}{j^s}-\int_j^{j+1}\frac{dx}{x^s}\right)
\label{6.4}
\end{equation}
and its alternating version
$$
\Upsilon^{*}(s):=\sum_{j=1}^{\infty}(-1)^{j+1}\left(\frac{1}{j^s}-\int_j^{j+1}\frac{dx}{x^s}\right)
$$
defined for $s\ge 0.$ The name of the function $\Upsilon(s)$ comes from the fact
that $\Upsilon(1)=\gamma$ and that the series $\sum_{j=1}^{\infty}1/j^s$ defines the Riemann
zeta function. Moreover, it is easily seen that $\Upsilon(1)=\gamma_{1,1}(1)$ and
$\Upsilon^{*}(1)=\gamma_{1,1}(-1).$ In \cite{la} it was shown that $\Upsilon(s)$ is infinitely
differentiable on ${\mathbb R}^{+}$ and its $k$-th derivative $\Upsilon^{(k)}(s)$ can be obtained
by termwise $k$-times differentiation of the series (\ref{6.4}):
\begin{equation}
\Upsilon^{(k)}(s)=(-1)^k\sum_{j=1}^{\infty}\left(\frac{\log^k j}{j^s}-
\int_{j}^{j+1}\frac{\log^k x}{x^s}\,dx\right).
\label{6.5}
\end{equation}
Setting $s=1$ in (\ref{6.5}) we get the following relation between the zeta-generalized Euler constant function
and Stieltjes constants (\ref{6.3}):
$$
\Upsilon^{(k)}(1)=(-1)^k \gamma_k.
$$
The formula (\ref{6.1}), as well as (\ref{6.4}), can be further generalized to
$$
\gamma_f=\lim_{n\to\infty}\left(\sum_{k=1}^nf(k)-\int_1^nf(x)\,dx\right)
$$
for some arbitrary positive decreasing function $f$ (see \cite{sa}).
For example, $f_n(x)=\frac{\log^n x}{x}$ gives rise to the Stieltjes constants, and
$f_s(x)=x^{-s}$ gives $\gamma_{{f_s}}=\frac{(s-1)\zeta(s)-1}{s-1},$
where again the limit (\ref{6.2}) appears.

There are other generalizations including a two-dimensional version of Euler's constant
and a lattice sum form. 
For a survey of further results and an extended bibliography, see 
\cite[Sections 1.5, 1.10, 2.21, 7.2]{fi}.





\section{Acknowledgements} 

We thank the Max Planck
Institute for Mathematics at Bonn where this research was carried
out. Special gratitude is due to professor B.~C.~Berndt for
providing paper \cite{bebo}. We thank the referee for useful comments, which improved
the presentation of the paper.

\begin{thebibliography}{99}

\bibitem{ad}
A.~W.~Addison,  A series representation for Euler's constant,
{\it Amer. Math. Monthly} {\bf 74} (1967), 823--824.
\bibitem{assh}
J.-P.~Allouche, J.~Shallit, J.~Sondow,  Summation of series
defined by counting blocks of digits, {\it J. Number Theory} {\bf 123}
(2007), no. 1, 133--143.
\bibitem{as}
J.-P.~Allouche, J.~Shallit,  Sums of digits and the Hurwitz
zeta function,  {\it Analytic Number Theory} (Tokyo, 1988).
Lecture Notes in Math., 1434, Springer, Berlin, 1990, pp. 19--30.
\bibitem{be}
B.~C.~Berndt,  On the Hurwitz zeta-function, {\it Rocky Mount.~J.~Math.}
{\bf 2} (1972), 151--157.
\bibitem{bebo}
B.~C.~Berndt, D.~C.~Bowman,  Ramanujan's short unpublished
manuscript on integrals and series related to Euler's constant,
in {\it Constructive, Experimental and Nonlinear Analysis} (Limoges,
1999),  CMS Conf. Proc., 27, Amer. Math. Soc., Providence,
RI, 2000, pp.\ 19--27.
\bibitem{br}
W.~E.~Briggs, The irrationality of $\gamma$ or sets of similar constants,
{\it Norske Vid. Selsk. Forh. (Trondheim)} {\bf 34} (1961), 25--28.
\bibitem{carl}
L.~Carlitz,  Advanced problem 5601, Solution by Heiko
Harborth, {\it Amer. Math. Monthly} {\bf 76} (1969), 567--568.
\bibitem{ca}
E.~Catalan,  Sur la constante d'Euler et la fonction de
Binet, {\it J. Math. Pures Appl.} {\bf 1} (1875), 209--240.
\bibitem{df}
H.~G.~Diamond, K.~Ford,  Generalized Euler constants, {\it Math. Proc. Camb.
Phil. Soc.} {\bf 145} (2008), 27--41.
\bibitem{dil}
K.~Dilcher,  Generalized Euler constants for arithmetical progressions,
{\it Math. Comp.} {\bf 59} (1992), 259--282.
\bibitem{fi}
S.~Finch, {\it Mathematical Constants,} Cambridge University
Press, Cambridge, 2003.
\bibitem{gs}
J.~Guillera, J.~Sondow,  Double integrals and infinite
products for some classical constants via analytic continuations
of Lerch's transcendent, {\it Ramanujan J.} {\bf 16} (2008), no. 3, 247--270.
\bibitem{ha}
G.~H.~Hardy,  Note on Dr.~Vacca's series for $\gamma,$ {\it Quart. J. Pure Appl. Math.}
{\bf 43} (1912), 215--216.
\bibitem{he0}
Kh.~Hessami Pilehrood, T.~Hessami Pilehrood,  Arithmetical properties
of some series with logarithmic coefficients, {\it Math. Z.} {\bf 255} (2007), 117--131.
\bibitem{ka}
S.~Kanemitsu,  On evaluation of certain limits in closed form, {\it Th\'eorie des Nombres,}
Proc. 1987 Qu\'ebec conf., ed.~J.-M.~De Koninck and C.~Levesque, Gruyter, 1989, pp.~459--474.
\bibitem{kl}
J.~C.Kluyver,  On certain series of Mr.~Hardy, {\it Quart. J. Pure Appl. Math.} {\bf 50}
(1927), 185--192.
\bibitem{kn}
J.~Knopfmacher,  Generalised Euler constants, {\it Proc. Edinburgh Math. Soc.} {\bf 21}
(1978), 25--32.
\bibitem{kre}
R.~Kreminski,  Newton-Cotes integration for approximating Stieltjes (generalized
Euler) constants, {\it Math. Comp.} {\bf 72}, no. 243, 1379--1397.
\bibitem{la}
V.~Lampret,  An accurate approximation of zeta-generalized-Euler-constant functions,
{\it Cent. Eur. J. Math.} {\bf 8} (2010), no. 3, 488--499.
\bibitem{l}
D.~H.~Lehmer,  Euler constants for arithmetical progressions, {\it Acta Arith.}
{\bf 27} (1975), 125--142.
\bibitem{ra}
S.~Ramanujan, {\it Notebooks (\rm{2} volumes),} Tata Institute of
Fundamental Research, Bombay, 1957.
\bibitem{sa}
J.~S\'andor,  On generalized Euler constants and Schl\"omilch-Lemonnier type
inequalities, {\it J. Math. Anal. Appl.} {\bf 328} (2007), 1336--1342.
\bibitem{oeis}
N.~J.~A.~Sloane,  {\it The on-line encyclopedia of integer
sequences} (2005),  published online at
http://www.research.att.com/~njas/sequences/
\bibitem{s} J.~Sondow,  New Vacca-type rational series for
Euler's constant and its "alternating" analog
$\log\frac{4}{\pi},$ {\it Additive Number Theory, Festschrift in Honor of the Sixtieth
Birthday of M.~B.~Nathanson} (D.~Chudnovsky and G.~Chudnovsky, eds.), Springer, 2010.
arXiv:0508042[math.NT]
\bibitem{sh}
J.~Sondow, P.~Hadjicostas,  The generalized-Euler-constant
function $\gamma(z)$ and a generalization of Somos's quadratic
recurrence constant, {\it J. Math. Anal. Appl.} {\bf 332} (2007), no.
1, 292--314.
\bibitem{st}
T.~J.~Stieltjes, {\it Correspondance d'Hermite et de Stieltjes,} V.1 and 2.
Gauthier-Villars, Paris, 1905.
\bibitem{ta}
T.~Tasaka,  Note on the generalized Euler constants,
{\it Math. J. Okayama Univ.} {\bf 36} (1994), 29--34.
\bibitem{va}
G.~Vacca,  A new series for the Eulerian constant
$\gamma=.577\cdots,$ {\it Quart. J. Pure Appl. Math.} {\bf 41}(1910),
363--364.
\end{thebibliography}


\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11Y60; Secondary 
65B10, 40A05, 05A15.

\noindent \emph{Keywords: } 
Euler constant, series summation, generating function,
Lerch transcendent, generalized Somos constants, Glaisher-Kinkelin constant.


\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences
\seqnum{A000120},
\seqnum{A023416}, and
\seqnum{A070939}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received October 19 2009;
revised version received June 27 2010. 
Published in {\it Journal of Integer Sequences}, July 9 2010.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


\end{document}

                                                                                


.
