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\begin{document}
%% \begin{linenumbers}

\markboth{W.\,Chu}{Logarithms of a Binomial series}

\title{Logarithms of a binomial series: Extension of a series of Knuth}

\author[W.\,Chu]{Wenchang Chu\corrauth}

\address{School of Mathematics and Statistics, Zhoukou Normal University, 466\,001 Henan, P.\,R.\,China}

\email{{\tt chu.wenchang@unisalento.it} (W.\,Chu)}

\begin{abstract}
The Lagrange expansion formula is employed to determine
the Maclaurin series for the logarithms of Lambert series
of binomial coefficients, extending the log--squared
of the Catalan generating function due to Knuth (2015).
\end{abstract}

\keywords{Catalan numbers, harmonic numbers, elementary symmetric function, Bell polynomial, Lagrange expansion formula}

\ams{Primary 05A15; Secondary 05A10}

\maketitle

\section{Introduction and motivation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\mb{N}$ be the set of natural numbers with
$\mb{N}_0=\{0\}\cup\mb{N}$. Catalan numbers
(cf.~\cite{koshy}) defined by
\pq{C_n=\frac1{n+1}\binm{2n}{n}
\quad\text{with}\quad n\in\mb{N}_0}
are one of the fascinating sequences
in classical combinatorics with more
than fifty significant combinatorial
interpretations~\cite{stanley,stanley-catalan}.
Their ordinary generating function is given
by
\[
C(x)=\sum_{n\ge0}C_nx^n=\frac{1-\sqrt{1-4x}}{2x}.
\]
Recently, Knuth~\cite{knuth} discovered the following
remarkable log--squared series
\pq{\log^2C(x)=\label{gf-2c}
\sum_{m\ge2}\frac{x^m}{m}
\binm{2m}{m}\big\{H_{2m-1}-H_m\big\}, }
where   harmonic numbers are defined by
\[
H_m:=H_m^{\sss\ang{1}}
\qquad\text{with}\qquad
H_m^{\sss\ang{n}}
=\sum_{k=1}^m\frac1{k^n}.
\]
The aim of this short paper is to extend Knuth's identity
to Lambert's binomial series (see~\cite{lam-a,lam-b})
with higher powers. The main theorem will be proved in the
next section by means of the Lagrange expansion formula.
Then the paper will end with a few applications to convolution
identities involving harmonic numbers.

In order to facilitate the subsequent references,
we record briefly the elementary symmetric functions
and their representation in terms of power sum functions.
For an indeterminate $y$ and $m,n\in\mb{N}$,
define the power sums
\[
\mc{P}_n(y|m):=\sum_{i=1}^m\frac{1}{(y+i)^{n}}
\]
and the elementary symmetric functions
\[
\Ome_{0}(y|m)=1
\quad\text{and}\quad
\Ome_{n}(y|m)
:=\qqdn\sum_{1\le k_1<k_2<\cdots<k_n\le m}
\prod_{i=1}^{n}\frac1{y+k_i}.
\]
Their generating function can be manipulated as follows:
\pnq{\sum_{n=0}^{m}x^n\Ome_{n}(y|m)
=&\prod_{k=1}^m\Big(1+\frac{x}{y+k}\Big)\\
=&\exp\bigg\{\sum_{k=1}^m\log\Big(1+\frac{x}{y+k}\Big)\bigg\}\\
=&\exp\bigg\{\sum_{k=1}^m\sum_{i=1}^{\infty}
\frac{(-1)^{i-1}x^i}{i(y+k)^{i}}\bigg\}\\
=&\exp\bigg\{\sum_{i=1}^{\infty}x^i
\frac{(-1)^{i-1}\mc{P}_i(y|m)}{i}\bigg\}.}
By extracting the coefficient of $x^n$ from the two extreme
members of these equations, we obtain the expression
in terms of power sums (cf. \cite[\S1.2]{mac}
and \cite{chu99a,chu06f,chu10f})
\pq{\Ome_{n}(y|m)
=\sum_{\sig(n)}
(-1)^n\prod_{i=1}^n
\frac{\mc{P}_i^{j_i}(y|m)}{j_i!}
\Big(\frac{-1}{i}\Big)^{j_i},}
where the multiple sum runs over all the partitions
$\sig(n)$ of $n$ indexed by nonnegative integers
$\{j_1,j_2,\cdots,j_n\}$ subject to the condition
$j_1+2j_2+\cdots+nj_n=n$. Denote further by $\sig_{k}(n)$
the subset of partitions of $n$ with exactly $k$
parts, characterized by $\{j_1,j_2,\cdots,j_n\}$
subject to $j_1+2j_2+\cdots+nj_n=n$ and
$j_1+j_2+\cdots+j_n=k$. Then the exponential
Bell polynomials (see \cite[\S3.3]{comtet})
read:
\[\mf{Y}_n(x_1,x_2,\cdots,x_n)
:=\sum_{k=0}^n
\mf{B}_{n,k}(x_1,x_2,\cdots,x_{n-k+1})\]
with the partial Bell polynomials   given
by $\mf{B}_{0,0}(x_1)=1$ and
\pnq{\mf{B}_{n,k}(x_1,x_2,\cdots,x_{n-k+1})
:=&\frac{n!}{k!}\sum_{\sig_{k}(n)}
\binm{k}{j_1,j_2,\cdots,j_n}
\prod_{i=1}^n\Big(\frac{x_i}{i!}\Big)^{j_i}\\
=&\frac{1}{k!}\sum\sbs{i_1+i_2+\cdots+i_k=n\\(i_j\in\mb{N})}
\binm{n}{i_1,i_2,\cdots,i_k}
\prod_{j=1}^kx_{i_j}.}

We have therefore the following alternative expression
of $\Ome_{n}(y|m)$ by the complete Bell polynomials:
\pq{\pp{&\Ome_{n}(y|m)
=\frac1{n!}\times
\mf{Y}_n(\mc{Q}_1,\mc{Q}_2,\cdots,\mc{Q}_n),\\
\text{where}\qquad
&\mc{Q}_k:=\mc{Q}_k(y|m)
=(-1)^{k-1}(k-1)!\mc{P}_k(y|m).}}

It may also be worth remarking that the last formula
for $\Ome_{n}$ is effectively a sum over the permutations
of $\{1,2,\cdots,n\}$. For example, there are 6 permutations
of $\{1,2,3,4\}$ consisting of a single cycle, 1 permutation
with four cycles of length 1, and $\binm42=6$ permutations
with cycle lengths (2,1,1). There are two different kinds
of permutations of $\{1,2,3,4\}$ with two cycles; 3 with
cycle lengths (2,2) and $2\binm43=8$ permutations with
cycle lengths (3,1). If we attach $(-1)^k$ to each
of these coefficients, where $k$ is the number of cycles,
and divide by the total number $4!=24$ of permutations,
then we get the formula for $\Ome_{4}$ in terms
of $\mc{P}_1,\mc{P}_2,\mc{P}_3,\mc{P}_4$.

The first five expressions are displayed below
with the common parameters $(y|m)$ being suppressed:
\[\pp{
\Ome_1=&\mc{P}_1,\\
\Ome_2=&\frac{1}{2}(\mc{P}_1^2-\mc{P}_2),\\
\Ome_3=&\frac{1}{6}(\mc{P}_1^3-3\mc{P}_1\mc{P}_2+2\mc{P}_3),\\
\Ome_4=&\frac{1}{24}(\mc{P}_1^4-6\mc{P}_1^2\mc{P}_2
	+8\mc{P}_1\mc{P}_3+3\mc{P}_2^2-6\mc{P}_4),\\
\Ome_5=&\frac{1}{120}
        (\mc{P}_1^5-10\mc{P}_1^3\mc{P}_2+20\mc{P}_1^2\mc{P}_3
	+15\mc{P}_1\mc{P}_2^2-30\mc{P}_1\mc{P}_4-20\mc{P}_2\mc{P}_3+24\mc{P}_5).}\]


\section{Main theorem and proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In classical analysis and enumerative combinatorics,
the Lagrange expansion formula is fundamental.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{lemma}[\text{Lagrange expansion formula \cite{lag}: see \cite[\S3.8]{comtet} and \cite{chu98b,chu13a}}]
For a formal power series  $\vph(x)$ subject to $\vph(0)\not=0$,
the functional equation $x=y/\vph(y)$ determines $y$
as an implicit function of $x$. Then for another formal
power series $F(y)$ in the variable $y$, the following
expansions hold for both composite series:
\pmq{
F(y(x))&=\label{lag-inv}F(0)+\sum_{n=1}^{\infty}
\frac{x^n}{n}[y^{n-1}]\big\{F'(y)\vph^n(y)\big\},\\
\frac{F(y(x))}{1-\big(y\vph'(y)/\vph(y)\big)}
&=\label{lag+inv}
\sum_{n=0}^{\infty}{x^n}
[y^{n}]\big\{F(y)\vph^n(y)\big\}.}
\end{lemma}

By taking $F(y)=(1+y)^{\alp}$ and $\vph(y)=(1+y)^{\lam}$
in this lemma, we recover the binomial series
due to Lambert \cite{lam-a,lam-b}
(see also \cite{chu98b,chu02a,gould}, \cite[\S5.4]{graham}
and \cite[\S5.4]{riordan}):
\[\left.\pp{
&\sum_{k\ge0}\frac{\alp}{\alp+k\lam}
\binm{\alp+k\lam}{k}x^k
=(1+y)^{\alp}\\
&\sum_{k\ge0}
\binm{\alp+k\lam}{k}x^k
=\frac{(1+y)^{\alp+1}}{1+y-\lam y}}
\right\}\quad\text{where}\quad
y:=y_{\lam}(x)=x(1+y)^{\lam}.\]
Taking $\alp=1$ now gives the generating function
\[C_{\lam}(x):=1+y_\lam(x)
=\sum_{n\ge0}\binm{1+n\lam}{n}
\frac{x^n}{1+n\lam}\]
of the extended Catalan numbers, where $y=x(1+y)^{\lam}$.
In particular, we have the generating function
$C(x)=1+y_2(x)$ for the usual Catalan numbers.

By making use of \eqref{lag-inv}, we can expand,
under the specifications $F(y)=\log(1+y)$ and
$\vph(y)$ as before, the logarithm function
\pq{\log C_{\lam}(x)=\label{log-1}
\sum_{m\ge1}\frac{x^m}{m\lam}\binm{m\lam}{m}.}
Furthermore, we can show the following main theorem.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{theorem} \label{log-n} Let $n \in\mb{N}$ and $\lam\in\mb{N}$. It holds:
\[
\log^nC_\lam(x)=n!\sum_{m\ge n}
\frac{x^m}{m\lam}\binm{m\lam}{m}
\Ome_{n-1}(m\lam-m|{m-1}).
\]
\end{theorem}

\proof
The formula in the theorem is clearly true for $n=1$
in view of \eqref{log-1}. Suppose that the formula
is valid for $n$. Then we have to prove it for $n+1$.
Because
\[
\log^nC_\lam(x)\times\log C_\lam(x)=\log^{n+1}C_\lam(x),
\]
the formula to be shown is equivalent to %% the following one
\pq{\sum_{k=n}^{m-1}
\frac{\Ome_{n-1}(k\lam-k|{k-1})}{k\lam}\binm{k\lam}{k}
\frac{\binm{m\lam-k\lam}{m-k}}{m\lam-k\lam}
=\label{check}
\frac{n+1}{m\lam}\binm{m\lam}{m}
\Ome_{n}(m\lam-m|{m-1}).}

Recall the Hagen--Rothe convolution formula
(cf. \cite{chu10b} and \cite{gould})
\[\sum_{k=0}^m\frac{x}{x+k\lam}\binm{x+k\lam}{k}
\frac{y}{y+m\lam-k\lam}\binm{y+m\lam-k\lam}{m-k}
=\frac{x+y}{x+y+m\lam}\binm{x+y+m\lam}{m}.\]
Moving the $k=m$ term to the other side, dividing by $y$,
and letting $y\to0$, we get
\pnq{\sum_{k=0}^{m-1}
\frac{x\binm{x+k\lam}{k}}{x+k\lam}
\frac{\binm{m\lam-k\lam}{m-k}}{m\lam-k\lam}
=&\lim_{y\to0}
\Bigg\{\frac{(x+y)\binm{x+y+m\lam}{m}}{y(x+y+m\lam)}
-\frac{x\binm{x+m\lam}{m}}{y(x+m\lam)}\Bigg\} }
%%[\\frac{\binm{x+m\lam}{m}}{x+m\lam}
%%\bigg\{1+x\Ome_1(x+m\lam-m|{m-1})\bigg\}\]
which can be reformulated equivalently as the equality
\pq{\sum_{k=0}^{m-1}
\frac{x\binm{x+k\lam}{k}}{x+k\lam}
\frac{\binm{m\lam-k\lam}{m-k}}{m\lam-k\lam}
=\label{b-rr}
\bigg\{1+x\frac{\text{d}}{\text{d}x}\bigg\}
\frac{\binm{x+m\lam}{m}}{x+m\lam}.}
Recalling the generating function
\[\sum_{n=0}^{m-1}x^n\Ome_{n}(y|{m-1})
=\prod_{k=1}^{m-1}\Big(1+\frac{x}{y+k}\Big)
=\frac{y+m}{x+y+m}\frac{\binm{x+y+m}{m}}{\binm{y+m}{m}}\]
we have
\[[x^n]\frac{\binm{x+y+m}{m}}{x+y+m}
=\frac{\binm{y+m}{m}}{y+m}\Ome_{n}(y|{m-1}),\]
where $[x^n]\psi(x)$ stands for the coefficient
of $x^n$ in the formal power series $\psi(x)$.
Then the coefficient of $x^n$ from the sum
on the left-hand side of \eqref{b-rr} reads:
\pnq{[x^n]\sum_{k=0}^{m-1}
\frac{x\binm{x+k\lam}{k}}{x+k\lam}
\frac{\binm{m\lam-k\lam}{m-k}}{m\lam-k\lam}
=&\sum_{k=0}^{m-1}
\frac{\binm{m\lam-k\lam}{m-k}}{m\lam-k\lam}
[x^{n-1}]\frac{\binm{x+k\lam}{k}}{x+k\lam}\\
=&\sum_{k=0}^{m-1}
\frac{\Ome_{n-1}(k\lam-k|{k-1})}{k\lam}\binm{k\lam}{k}
\frac{\binm{m\lam-k\lam}{m-k}}{m\lam-k\lam}.}
The coefficient of $x^n$ from the right-hand side member
of \eqref{b-rr} can be extracted similarly��
\pnq{[x^n]\bigg\{1+x\frac{\text{d}}{\text{d}x}\bigg\}
\frac{\binm{x+m\lam}{m}}{x+m\lam}
=&[x^n]\frac{\binm{x+m\lam}{m}}{x+m\lam}
+n[x^{n}]\frac{\binm{x+m\lam}{m}}{x+m\lam}\\
=&\frac{n+1}{m\lam}\binm{m\lam}{m}\Ome_{n}(m\lam-m|{m-1}).}
Therefore \eqref{check} is verified and Theorem~\ref{log-n}
follows by induction.
\endproof


\section{Examples as applications}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


By making use of the relation
\[
\mc{P}_n(y|m)
=H_{m+y}^{\sss\ang{n}}
-H_{y}^{\sss\ang{n}}
\quad\text{for}\quad
y\in\mb{N}_0,
\]
from Theorem~\ref{log-n} we can derive
the following infinite series expressions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition} \label{log<5} Let $\lam\in\mb{N}$. It holds:
\pnq{
\log^2C_\lam(x)=2\sum_{m\ge2}
\frac{x^m}{m\lam}\binm{m\lam}{m}
&\big\{H_{m\lam-1}-H_{m\lam-m}\big\},\\
\log^3C_\lam(x)=3\sum_{m\ge3}
\frac{x^m}{m\lam}\binm{m\lam}{m}
&\Big\{\big(H_{m\lam-1}-H_{m\lam-m}\big)^2
-\big(H^{\sss\ang{2}}_{m\lam-1}
-H^{\sss\ang{2}}_{m\lam-m}\big)\Big\},\\
\log^4C_\lam(x)=4\sum_{m\ge4}
\frac{x^m}{m\lam}\binm{m\lam}{m}
&\Bigg\{\mult{r}{
(H_{m\lam-1}-H_{m\lam-m}\big)^3
+2\big(H^{\sss\ang{3}}_{m\lam-1}
-H^{\sss\ang{3}}_{m\lam-m}\big)\\[2mm]
-3\big(H_{m\lam-1}-H_{m\lam-m}\big)
\big(H^{\sss\ang{2}}_{m\lam-1}
-H^{\sss\ang{2}}_{m\lam-m}\big)}
\Bigg\}.}
\end{proposition}

In particular, we recover, for $\lam=2$, Knuth's series
\eqref{gf-2c} and two further  series of cubic and quartic
logarithms of the Catalan generating function in terms
of generalized harmonic numbers.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}\label{log>5}
\pnq{
\log^3C(x)=3\sum_{m\ge3}
\frac{x^m}{2m}\binm{2m}{m}
&\Big\{\big(H_{2m-1}-H_{m}\big)^2
-\big(H^{\sss\ang{2}}_{2m-1}
-H^{\sss\ang{2}}_{m}\big)\Big\},\\
\log^4C(x)=4\sum_{m\ge4}
\frac{x^m}{2m}\binm{2m}{m}
&\Bigg\{\mult{r}{
(H_{2m-1}-H_{m}\big)^3
+2\big(H^{\sss\ang{3}}_{2m-1}
-H^{\sss\ang{3}}_{m}\big)\\[2mm]
-3\big(H_{2m-1}-H_{m}\big)
\big(H^{\sss\ang{2}}_{2m-1}
-H^{\sss\ang{2}}_{m}\big)}
\Bigg\}.}
\end{corollary}

By applying the exponential law to Theorem~\ref{log-n},
we get the next convolution formula.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}For $\alp,\gam\in\mb{N}$ and $m\in\mb{N}$ it holds:
\pnq{\sum_{k=\alp}^{m-\gam}\binm{k\lam}{k}&
\frac{\Ome_{\alp-1}(k\lam-k|{k-1})}{k\lam}
\binm{m\lam-k\lam}{m-k}
\frac{\Ome_{\gam-1}((m-k)(\lam-1)|{m-k-1})}{(m-k)\lam}\\
=&\binm{\alp+\gam}{\alp}\binm{m\lam}{m}
\frac{\Ome_{\alp+\gam-1}(m\lam-m|{m-1})}{m\lam}.}
\end{proposition}

When $\alp=\gam=1$ and $\alp=2,\:\gam=1$, the above formula
reduces to the following two further interesting ones.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}Let $\lam\in\mb{N}$ and $m\in\mb{N}$. It holds:
\pnq{\sum_{k=1}^{m-1}\binm{k\lam}{k}
&\binm{m\lam-k\lam}{m-k}
\frac{m}{k(m-k)}
=2\lam\binm{m\lam}{m}
\big\{H_{m\lam-1}-H_{m\lam-m}\big\},\\
\sum_{k=2}^{m-1}\binm{k\lam}{k}
&\binm{m\lam-k\lam}{m-k}
\frac{H_{k\lam-1}-H_{k\lam-k}}{k(m-k)}\\
=&\frac{3\lam}{2m}\binm{m\lam}{m}
\Big\{\big(H_{m\lam-1}-H_{m\lam-m}\big)^2
-\big(H^{\sss\ang{2}}_{m\lam-1}-H^{\sss\ang{2}}_{m\lam-m}\big)\Big\}.}
\end{corollary}

In addition, by combining \eqref{log-1}
with Theorem~\ref{log-n} we can derive the following multiple
convolution identity.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{corollary}Let $\lam\in\mb{N}$ and $m,n\in\mb{N}$.
Denote by $h_k(\lam)$ the falling factorial
\[h_k(\lam)=(k\lam-1)(k\lam-2)\cdots(k\lam-k+1)
\quad\text{for}\quad k\ge1.\]
Then there holds the summation formula
\[h_m(\lam)\Ome_{n-1}(m\lam-m|{m-1})
=\mf{B}_{m,n}\big(h_1(\lam),h_2(\lam),\cdots,h_{m-n+1}(\lam)\big).\]
\end{corollary}

Analogously, there are the series for Abel coefficients~\cite{abel}
(see also \cite{chu98b,chu02a} for example)
\[\left.\pp{
&\sum_{k\ge0}\frac{\alp}{\alp+k\lam}
\frac{(\alp+k\lam)^k}{k!}x^k
=e^{z\alp}\\
&\sum_{k\ge0}
\frac{(\alp+k\lam)^k}{k!}x^k
=\frac{e^{z\alp}}{1-z\lam}}
\right\}\quad\text{where}\quad
z:=z_{\lam}(x)=xe^{z\lam}.\]

In this case, taking into account \eqref{lag-inv} we have the expressions
\[z_{\lam}(x)=\label{exp-1}
\sum_{m\ge1}\frac{(m\lam)^{m-1}}{m!}{x^m}
\quad\text{and}\quad
z^n_{\lam}(x)=%\label{exp-1}
\sum_{m\ge n}\frac{n}{m}
\frac{(m\lam)^{m-n}}{(m-n)!}{x^m}.\]
They lead to the following multiple convolution formula
for the number of forests of $n$ labeled rooted trees on $m$ vertices
\[\binm{m-1}{n-1}m^{m-n}
=\mf{B}_{m,n}\big(1^0,2^1,3^2,\cdots,(m-n+1)^{m-n}\big)\]
%\[\boxed{\sum\sbs{k_1+k_2+\cdots+k_n=m\\
%	k_\ii\ge1 (1\le\ii\le n)}
%\quad\prod_{\ii=1}^n
%\frac{k_{\ii}^{k_{\ii}-1}}{k_{\ii}!}
%=\frac{n}{m}\frac{m^{m-n}}{(m-n)!}}\]
as well as the convolution identity
\[
\sum_{k=\alp}^{m-\gam}\frac{\alp}{k}
\frac{k^{k-\alp}}{(k-\alp)!}
\frac{\gam}{m-k}
\frac{(m-k)^{m-k-\gam}}{(m-k-\gam)!}
=\frac{\alp+\gam}{m}
\frac{m^{m-\alp-\gam}}{(m-\alp-\gam)!},
\]
which can also be derived from the Abel identity
(see \cite{chu10b} and \cite[\S3.1]{comtet}).

\section*{Acknowledgement}
The author is grateful to an anonymous
referee for the careful reading, critical comments and
valuable suggestions that make the manuscript improved
substantially in revision.


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