Another Identity for Complete Bell Polynomials based on Ramanujan's Congruences

TL;DR

This paper introduces a novel identity for complete Bell polynomials derived from Ramanujan's generating function for specific partition numbers, enriching the mathematical understanding of partition functions and Bell polynomials.

Contribution

The paper presents a new identity linking complete Bell polynomials to Ramanujan's congruences for partition numbers, offering a fresh perspective in combinatorial analysis.

Findings

Derived a new identity for complete Bell polynomials

Connected Bell polynomials with Ramanujan's partition congruences

Enhanced understanding of partition functions through Bell polynomial identities

Abstract

Let $p(n)$ be the number of partition 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.

Equations

$$B_{n,k}(x_1,x_2,\dots ,x_{n-k+1})$$

$$j_1+j_2+\cdots +j_{n-k+1}$$

$$j_1+2j_2+\cdots +(n-k+1)j_{n-k+1}$$

$$B_n(x_1\dots x_n)$$

$$\displaystyle\text{exp}\Big{(}\sum_{m=1}^{\infty}x_{m}\frac{t^{m}}{m!}\Big{)}$$

$$(a;q{)}_n:=\prod _{k=0}^{n-1}(1-a{q}^k)=(1-a)(1-aq)(1-a{q}^2)\dots (1-a{q}^{n-1}).$$

$$(a;q{)}_{\infty }:=\prod _{k=0}^{\infty }(1-a{q}^k).$$

$$\sum _{n=0}^{\infty }p(n)q^n$$

$$p(5k+4)\equiv 0\text{ (mod 5)};p(7k+5)\equiv 0\text{ (mod 7)};p(11k+6)\equiv 0\text{ (mod 11)}.$$

$$\sum _{k=0}^{\infty }p(5k+4)q^k$$

$$\sum _{k=0}^{\infty }p(7k+5)x^k$$

$$\sigma (mn)$$

$$\sigma (p^m)$$

$$\sigma (n)$$

$$\displaystyle\sigma\Big{(}\frac{n}{7}\Big{)}$$

$$d_n$$

$$e_n$$

$$7B_n(1!d_1,2!d_2,\dots ,n!d_n)+49n{B}_{n-1}(1!e_1,2!e_2,\dots ,(n-1)!e_{n-1})$$

$$G(x)$$

$$H(x)$$

$$\ln (G(x))$$

$$\ln (H(x))$$

$$\ln (G(x))$$

$$\ln (H(x))$$

$$f_1(x)$$

$$f_2(x)$$

$$f_1(x)$$

$$a_i$$

$$f_2(x)$$

$$b_i$$

$$\ln (G(x))$$

$$d_i$$

Full text

1 Introduction

Let $(x_{1},x_{2},\dots)$ be a sequence of real numbers. The partial exponential Bell polynomials are polynomials given by

[TABLE]

where $\pi(n,k)$ is the positive integer sequence $(j_{1},j_{2},j_{n-k+1})$ satisfies the following equations:

[TABLE]

For $n\geq 1$, the $n^{\text{th}}$-complete exponential Bell polynomial $B_{n}(x_{1},\dots,x_{n})$ is the following:

[TABLE]

The complete exponential Bell polynomials can also be defined by power series expansion as follows:

[TABLE]

where $B_{0}\equiv 1$. Bell polynomials were first introduced by Bell [2]. The books written by Comtet [4] and Riordan [6] 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,

[TABLE]

Considered as a formal power series in $q$, the definition of $q$-Pochhammer symbol can be extended to an infinite product. That is,

[TABLE]

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

[TABLE]

Andrew’s book [1] serves as an excellent reference to the theory of partitions.

Ramanujan’s congruences are congruence properties for $p(n)$:

[TABLE]

In 1919, Ramanujan [5] proved the first two congruences by the following two identities:

[TABLE]

Bouroubi and Benyahia-Tani [3] proved an identity for complete Bell polynomials based on (2). As an analogue to their result, we give an identity for complete Bell polynomials based on (3). In other words, we derive formulas that relate $p(7n+5)$ and certain complete Bell polynomials.

2 Main Theorem

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

[TABLE]

If $m\geq 1$, then

[TABLE]

Lemma 1**.**

Let $n=7^{m}n^{\prime}$ such that $m\geq 1$ and $\text{g.c.d.}(n,n^{\prime})=1$. Then

[TABLE]

Proof.

By (4) and (5),

[TABLE]

By a combination of (6) and (7), we get the desired result. ∎

Theorem 2**.**

Let $n\geq 1$. We write $n=7^{m}n^{\prime}$ where $m\geq 0$ and $\text{g.c.d.}(n,n^{\prime})=1$. Let $d_{n}$ and $e_{n}$ be the following sequences of numbers respectively:

[TABLE]

Then we have the following identity:

[TABLE]

Proof.

Let $G(x)$ and $H(x)$ be the following functions:

[TABLE]

$G(x)$ 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 (8) and (9),

[TABLE]

By using the power series expansion of $\ln(1-x)$, we get

[TABLE]

Let $f_{1}(x)$ and $f_{2}(x)$ be the following two functions:

[TABLE]

$f_{1}(x)$ has non-zero coefficients for $x^{m}$ if and only if $m$ is a multiple of $7$. More precisely, let

[TABLE]

Then,

[TABLE]

We write $f_{2}(x)$ as

[TABLE]

where

[TABLE]

By (10), (12), (13), (14), (15), we get

[TABLE]

where

[TABLE]

By Lemma 1, we write (17) as

[TABLE]

for $i=7^{m}i^{\prime}$, $m\geq 0$. Similarly, by (11), (12), (13), (14), (15), we get

[TABLE]

where

[TABLE]

By Lemma 1, we write (20) as

[TABLE]

for $i=7^{m}i^{\prime}$, $m\geq 0$. By (1), (16), (18), we have

[TABLE]

By (1), (19), (21), we have

[TABLE]

By (3), (8), (9), (22), (23), we have the following identity:

[TABLE]

as desired. ∎

3 Acknowledgement

The author is grateful to Victor Bovdi for pointing out some references about this topic. The author is also grateful to Mohamed El Bachraoui for some valuable discussions on the topic. The author is supported by Startup-Grant-2016 (fund no. 31S263) from United Arab Emirates University.