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\begin{center}
\vskip 1cm{\LARGE\bf
Another Identity for Complete Bell  \\
Polynomials Based on Ramanujan's \\
\vskip .07in
Congruences
}
\vskip 1cm
\large
Ho-Hon Leung\\
Department of Mathematical Sciences \\
United Arab Emirates University \\
Al Ain, 15551\\
United Arab Emirates\\
\href{mailto:hohon.leung@uaeu.ac.ae}{\tt hohon.leung@uaeu.ac.ae}
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\begin{abstract}
Let $p(n)$ be the number of partitions of a positive integer $n$. We derive a new identity for complete Bell polynomials based on a generating function of $p(7n+5)$ given by Ramanujan.
\end{abstract}

\section{Introduction}  \label{section1}

Let $(x_1, x_2, \ldots)$ be a sequence of real numbers. The partial exponential Bell polynomials are polynomials given by 
\begin{align*}
B_{n,k}(x_1,x_2,\dots,x_{n-k+1}) &= \sum_{\pi(n,k)} \frac{n!}{j_1 ! j_2 ! \cdots j_{n-k+1}!} \Big(\frac{x_1}{1!} \Big)^{j_1} \Big(\frac{x_2}{2!} \Big)^{j_2}\cdots \Big(\frac{x_{n-k+1}}{(n-k+1)!} \Big)^{j_{n-k+1}}
\end{align*}where $\pi (n,k)$ is the positive integer sequence $(j_1, j_2,\dots, j_{n-k+1})$ satisfying the following equations: 
\begin{align*}
j_1 +j_2 +\dots +j_{n-k+1} &=k, \\
j_1 + 2 j_2 + \dots + (n-k+1) j_{n-k+1} &=n.
\end{align*}For $n\geq 1$, the $n^{\text{th}}$-complete exponential Bell polynomial $B_n(x_1, \dots, x_n)$ is as follows:
\begin{align*}
  B_n (x_1,\dots, x_n)  &=\sum_{k=1}^n B_{n,k} (x_1, \dots , x_{n-k+1}).
\end{align*}The complete exponential Bell polynomials can also be defined by power series expansion as follows:
\begin{align}
  \label{equation1}  \text{exp}\Big( \sum_{m=1}^\infty x_m \frac{t^m}{m!} \Big) &= \sum_{n=0}^\infty B_n (x_1, \dots , x_n) \frac{t^n}{n!},
\end{align}where $B_0\equiv 1$. Bell polynomials were first introduced by Bell \cite{Bell}. The books written by Comtet \cite{Comtet} and Riordan \cite{Riordan} serve as excellent references for the numerous applications of Bell polynomials in combinatorics. 

Let $(a;q)_n$ be the $q$-Pochhammer symbol for $n\geq 1$. That is, 
\[ (a;q)_n:=\prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots (1-aq^{n-1}).\]Considered as a formal power series in $q$, the definition of $q$-Pochhammer symbol can be extended to an infinite product. That is, \[(a;q)_\infty := \prod_{k=0}^\infty (1-aq^k).\]We note that $(q;q)_\infty$ is the Euler's function. Let $p(n)$ be the number of partitions of $n$. The generating function of $p(n)$ can be written as 
\begin{align*}
 \sum_{n=0}^\infty p(n) q^n &= \frac{1}{(q;q)_\infty}.
\end{align*}Andrew's book \cite{Andrews} serves as an excellent reference to the theory of partitions. 

Ramanujan's congruences are congruence properties for $p(n)$: 
\[ p(5k+4) \equiv 0  \text{ (mod $5$)};  \quad p(7k+5)\equiv 0 \text{ (mod $7$)}; \quad p(11k+6)\equiv 0 \text{ (mod $11$)}.\]In 1919, Ramanujan \cite{Ramanujan} proved the first two congruences by the following two identities: 
\begin{align}
\label{equation2}\sum_{k=0}^\infty p(5k+4) q^k &= 5 \frac{(q^5;q^5)_{\infty}^5}{(q;q)_{\infty}^6}, \\
\label{equation3}\sum_{k=0}^\infty p(7k+5) x^k &= 7\frac{(q^7;q^7)_{\infty}^3}{(q;q)_{\infty}^4}+ 49x \frac{(q^7;q^7)_{\infty}^7}{(q;q)_{\infty}^8}.
\end{align}

Bouroubi and Benyahia-Tani \cite{Sadek} proved an identity for complete Bell polynomials based on (\ref{equation2}). As an analogue to their result, we give an identity for complete Bell polynomials based on (\ref{equation3}). In other words, we derive formulas that relate $p(7n+5)$ and certain complete Bell polynomials.

\section{Main theorem} \label{section2}

Let $\sigma(n)$ be the sum of divisors (including $1$ and $n$) for $n$. It is well known that $\sigma(n)$ is a multiplicative function. That is, if $n$ and $m$ are coprime, then
\begin{align}
\label{equation4} \sigma(mn)&=\sigma(m) \sigma(n).
\end{align}If $m\geq 1$, then 
\begin{align}
\label{equation5} \sigma(p^m)&=1+p+\dots +p^{m-1}+p^m=\frac{p^{m+1}-1}{p-1}.
\end{align}

\begin{lemma} \label{lemma1}
Let $n=7^m n'$ such that $m\geq 1$ and $\gcd(n,n')=1$. Then,
\begin{align*}
\sigma(n) &= \frac{7^{m+1}-1}{7^m -1} \sigma\big( \frac{n}{7}\big).
\end{align*}
\end{lemma}

\begin{proof}By (\ref{equation4}) and (\ref{equation5}),
\begin{align}
\label{equation6} \sigma(n) &= \sigma (7^m) \sigma(n')=\frac{7^{m+1}-1}{6} \sigma\Big(\frac{n}{7^m}\Big), \\
\label{equation7} \sigma\Big(\frac{n}{7} \Big) &= \sigma \Big( 7^{m-1} \Big) \sigma\Big(\frac{n}{7^m} \Big) =\frac{7^m -1}{6}\sigma\Big( \frac{n}{7^m}\Big).
\end{align}By a combination of (\ref{equation6}) and (\ref{equation7}), we get the desired result.
\end{proof}

\begin{theorem}
Let $n\geq 1$. We write $n=7^m n'$ where $m\geq 0$ and $\gcd(n,n')=1$. Let $d_n$ and $e_n$ be the following sequences of numbers respectively:
\begin{align*}
d_n &= \frac{\sigma(n)}{n} \Big(1+\frac{18}{7^{m+1}-1}\Big),\\
e_n &= \frac{\sigma(n)}{n}\Big( 1+\frac{42}{7^{m+1}-1} \Big).
\end{align*} Then we have the following identity: 
\begin{align*}
 7B_n(1!d_1, 2!d_2,\dots, n!d_n)+49n B_{n-1}(1!e_1, 2!e_2 , \dots, (n-1)!e_{n-1}) &= n! p(7n+5).
\end{align*}
\end{theorem}

\begin{proof}
Let $G(x)$ and $H(x)$ be the following functions:
\begin{align}
\label{equation8} G(x) &= 7 \frac{(x^7 ; x^7)_\infty^3}{(x;x)_\infty^4}, \\
\label{equation9} H(x) &= 49x \frac{(x^7;x^7)_\infty^7}{(x;x)_\infty^8}.
\end{align}The functions $G(x)$ and $H(x)$ are well-defined on the interior of the unit disk in the complex plane by analytic continuation. We get the following two equations by (\ref{equation8}) and (\ref{equation9}), 
\begin{align*}
\ln (G(x)) &= \ln 7 +3 \sum_{i=1}^\infty \ln(1-x^{7i}) -4 \sum_{i=1}^\infty \ln (1-x^i), \\
 \ln (H(x)) &= \ln 49 +\ln x +7 \sum_{i=1}^\infty \ln (1-x^{7i}) -8 \sum_{i=1}^\infty \ln (1-x^i).
\end{align*}By using the power series expansion of $\ln (1-x)$, we get 
\begin{align}
\label{equation10} \ln (G(x)) &= \ln 7-3 \sum_{i=1}^\infty \sum_{j=1}^\infty \frac{x^{7ij}}{j} +4\sum_{i=1}^\infty \sum_{j=1}^\infty \frac{x^{ij}}{j}, \\
\label{equation11} \ln (H(x)) &= \ln 49 +\ln x -7\sum_{i=1}^\infty \sum_{j=1}^\infty \frac{x^{7ij}}{j} +8 \sum_{i=1}^\infty \sum_{j=1}^\infty \frac{x^{ij}}{j}.
\end{align}Let $f_1(x)$ and $f_2(x)$ be the following two functions: 
\begin{align}
\label{equation12}  f_1(x) &= \sum_{i=1}^\infty \sum_{j=1}^\infty \frac{x^{7ij}}{j}, \\
\label{equation13} f_2(x) &= \sum_{i=1}^\infty \sum_{j=1}^\infty \frac{x^{ij}}{j}.
\end{align}The function $f_1(x)$ has non-zero coefficients for $x^m$ if and only if $m$ is a multiple of $7$. More precisely, let 
\begin{align*}
f_1(x) &= \sum_{i=1}^\infty a_i x^i.
\end{align*}Then, 
\begin{align}
 \label{equation14} a_i &=
\begin{cases}
    \frac{\sigma(i/7)}{i/7},& \text{if }  7| i; \\
    0,              & \text{otherwise.}
\end{cases}
\end{align}We write the function $f_2(x)$ as 
\begin{align*}
  f_2(x) &= \sum_{i=1}^\infty b_i x^i
\end{align*}where 
\begin{align}
 \label{equation15} b_i &= \frac{\sigma(i)}{i}.
\end{align}By (\ref{equation10}), (\ref{equation12}), (\ref{equation13}), (\ref{equation14}), (\ref{equation15}), we get 
\begin{align}
\label{equation16} \ln (G(x))  &=\ln 7 +\sum_{i=1}^\infty d_i x^i 
\end{align}where 
\begin{align}
\label{equation17}  d_i &= 
\begin{cases}
    \frac{4\sigma(i)}{i}-\frac{3\sigma(i/7)}{i/7},& \text{if } 7|i;\\
    \frac{4\sigma(i)}{i},              & \text{otherwise.}
\end{cases}
\end{align}By Lemma \ref{lemma1}, we write (\ref{equation17}) as 
\begin{align}
\label{equation18} d_i &= \frac{\sigma(i)}{i} \Big( 1+\frac{18}{7^{m+1}-1} \Big)
\end{align}for $i=7^m i'$, $m\geq 0$. Similarly, by (\ref{equation11}), (\ref{equation12}), (\ref{equation13}), (\ref{equation14}), (\ref{equation15}), we get
\begin{align}
\label{equation19} \ln(H(x)) &= \ln 49 +\ln x + \sum_{i=1}^\infty e_i x^i
\end{align}where
\begin{align}
\label{equation20} e_i &= 
\begin{cases}
    \frac{8\sigma(i)}{i}-\frac{7\sigma(i/7)}{i/7},& \text{if } 7|i;\\
    \frac{8\sigma(i)}{i},              & \text{otherwise.}
\end{cases}
\end{align}By Lemma \ref{lemma1}, we write (\ref{equation20}) as 
\begin{align}
\label{equation21} e_i &= \frac{\sigma(i)}{i} \Big( 1+\frac{42}{7^{m+1}-1} \Big)
\end{align}for $i=7^m i'$, $m\geq 0$. By (\ref{equation1}), (\ref{equation16}), (\ref{equation18}), we have
\begin{align}
\nonumber G(x)&=\exp(\ln G(x)) =7\exp \Big(\sum_{n=1}^\infty d_n x^n  \Big)=7\exp\Big(\sum_{n=1}^\infty (n! d_n)\frac{x^n}{n!} \Big)  \\
\label{equation22}  &= 7\Big(\sum_{n=0}^\infty B_n (1! d_1, 2! d_2, \dots, n! d_n)\frac{x^n}{n!}\Big).
\end{align}By (\ref{equation1}), (\ref{equation19}), (\ref{equation21}), we have
\begin{align}
\nonumber H(x) &= \exp(\ln H(x)) = 49x \exp \Big(\sum_{n=1}^\infty e_n x^n \Big)=49x\exp \Big( \sum_{n=1}^\infty (n! e_n)\frac{x^n}{n!} \Big) \\
\nonumber  &= 49x \Big( \sum_{n=0}^\infty B_n (1!e_1, 2!e_2, \dots, n!e_n)\frac{x^n}{n!} \Big) \\
\label{equation23}  &= 49 \Big(\sum_{n=1}^\infty B_{n-1}(1!e_1, 2!e_2,\dots, (n-1)! e_{n-1})\frac{x^n}{(n-1)!}  \Big).
\end{align}By (\ref{equation3}), (\ref{equation8}), (\ref{equation9}), (\ref{equation22}), (\ref{equation23}), we have the following identity:
\begin{align}
\nonumber \sum_{n=1}^\infty p(7n+5) x^n&= \sum_{n=1}^\infty \Big( 7B_n(1!d_1, 2!d_2,\dots, n!d_n)+49n B_{n-1} (1!e_1, 2!e_2, \dots, (n-1)! e_{n-1})     \Big) \frac{x^n}{n!} 
\end{align}as desired.
\end{proof}


\section{Acknowledgments}
The author is grateful to the editor-in-chief and to the referee(s) for
carefully reading the paper. Their comments were helpful to improve the
quality of the article. The author is also grateful to Victor Bovdi for
pointing out some appropriate references, and thanks Mohamed El
Bachraoui for some valuable discussions on the topic. The author is
supported by Startup Grant 2016 (G00002235) from United Arab Emirates
University.






\begin{thebibliography}{9}


\bibitem{Andrews} G.~Andrews, \emph{The Theory of Partitions}, Cambridge University Press, 1976.

\bibitem{Bell} E.~T.~Bell, Exponential polynomials, \emph{Ann. Math.} {\bf 35} (1934), 258--277.

\bibitem{Sadek} S.~Bouroubi and N.~Benyahia~Tani, A new identity for
complete Bell polynomials based on a formula of Ramanujan, 
\emph{J. Integer Sequences} {\bf 12} (2009),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL12/Bouroubi/bouroubi25.html}{Article
09.3.5}.

\bibitem{Comtet} L.~Comtet, \emph{Advanced Combinatorics: The Art of Finite and Infinite Expansions}, D. Reidel Publishing Co., 1974.

\bibitem{Ramanujan} S.~Ramanujan, Congruence properties of partitions, \emph{Math. Z.} {\bf 9} (1921), 147--153.

\bibitem{Riordan} J.~Riordan, \emph{Combinatorial Identities}, Robert E. Krieger Publishing Co., 1979.



\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}: 
Primary 05A17; Secondary 11P81.

\noindent \emph{Keywords: } 
Bell polynomial, integer partition, divisor sum, Ramanujan's congruence.

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\noindent (Concerned with sequences \seqnum{A000041} and \seqnum{A071746}.)

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\vspace*{+.1in}
\noindent
Received  February 24 2018;
revised versions received  June 2 2018; June 3 2018.
Published in {\it Journal of Integer Sequences},
August 22 2018.

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