Another Identity for Complete Bell Polynomials Based on Ramanujan’s Congruences

Ho-Hon Leung Department of Mathematical Sciences United Arab Emirates University Al Ain, 15551 United Arab Emirates

                    hohon.leung@uaeu.ac.ae

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.

1 Introduction

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

B_{n,k}(x_{1}, x_{2}, …, x_{n−k+1}) = ∑_{π(n,k)} \frac{n!}{j_{1}!j_{2}! ⋯ j_{n−k+1}!}(\frac{x_{1}}{1!} )^{j_{1}}(\frac{x_{2}}{2!} )^{j_{2}} ⋯ (\frac{x_{n−k+1}}{(n − k + 1)!})^{j_{n−k+1}}

where π(n, k) is the positive integer sequence (j_{1}, j_{2}, …, j_{n−k+1}) satisfying the following equations:

j_{1} + j_{2} + ⋯ + j_{n−k+1} = k,
j_{1} + 2j_{2} + ⋯ + (n − k + 1)j_{n−k+1} = n.

For n ≥ 1, the n^{th}-complete exponential Bell polynomial B_{n}(x_{1}, …, x_{n}) is as follows:

B_{n}(x_{1}, …, x_{n}) = ∑_{k=1}^{n} B_{n,k}(x_{1}, …, x_{n−k+1}).

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

exp( ∑_{m=1}^{∞} x_{m} \frac{t^{m}}{m!}) = ∑_{n=0}^{∞} B_{n}(x_{1}, …, x_{n})\frac{t^{n}}{n!},  (1)

where B_{0} ≡ 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 ≥ 1. That is,

(a; q)_{n} := ∏_{k=0}^{n−1}(1 − aq^{k}) = (1 − a)(1 − aq)(1 − aq^{2}) ⋯ (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)_{∞} := ∏_{k=0}^{∞}(1 − aq^{k}).

We note that (q; q)_{∞} 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

∑_{n=0}^{∞} p(n)q^{n} = \frac{1}{(q; q)_{∞}} .

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

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

p(5k + 4) ≡ 0 (mod 5); p(7k + 5) ≡ 0 (mod 7); p(11k + 6) ≡ 0 (mod 11).

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

∑_{k=0}^{∞} p(5k + 4)q^{k} = 5\frac{(q^{5}; q^{5})_{∞}^{5}}{(q; q)_{∞}^{6}} ,  (2)

∑_{k=0}^{∞} p(7k + 5)x^{k} = 7\frac{(q^{7}; q^{7})_{∞}^{3}}{(q; q)_{∞}^{4}} + 49x\frac{(q^{7}; q^{7})_{∞}^{7}}{(q; q)_{∞}^{8}} .  (3)

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 σ(n) be the sum of divisors (including 1 and n) for n. It is well known that σ(n) is a multiplicative function. That is, if n and m are coprime, then

σ(mn) = σ(m)σ(n).  (4)

If m ≥ 1, then

σ(p^{m}) = 1 + p + ⋯ + p^{m−1} + p^{m} = \frac{p^{m+1} − 1}{p − 1}.  (5)

Lemma 1. Let n = 7^{m}n′ such that m ≥ 1 and gcd(n, n′) = 1. Then,

σ(n) = \frac{7^{m+1} − 1}{7^{m} − 1}σ(\frac{n}{7}).

Proof. By (4) and (5),

σ(n) = σ(7^{m})σ(n′) = \frac{7^{m+1} − 1}{6}σ(\frac{n}{7^{m}} ),  (6)
σ(\frac{n}{7}) = σ(7^{m−1})σ(\frac{n}{7^{m}} ) = \frac{7^{m} − 1}{6}σ(\frac{n}{7^{m}} ).  (7)

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

Theorem 2. Let n ≥ 1. We write n = 7^{m}n′ where m ≥ 0 and gcd(n, n′) = 1. Let d_{n} and e_{n} be the following sequences of numbers respectively:

d_{n} = \frac{σ(n)}{n}(1 + \frac{18}{7^{m+1} − 1}),
e_{n} = \frac{σ(n)}{n}(1 + \frac{42}{7^{m+1} − 1}).

Then we have the following identity:

7B_{n}(1!d_{1}, 2!d_{2}, …, n!d_{n}) + 49nB_{n−1}(1!e_{1}, 2!e_{2}, …, (n − 1)!e_{n−1}) = n!p(7n + 5).

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

G(x) = 7\frac{(x^{7}; x^{7})_{∞}^{3}}{(x; x)_{∞}^{4}} ,  (8)
H(x) = 49x\frac{(x^{7}; x^{7})_{∞}^{7}}{(x; x)_{∞}^{8}} .  (9)

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 (8) and (9),

ln(G(x)) = ln 7 + 3 ∑_{i=1}^{∞} ln(1 − x^{7i}) − 4 ∑_{i=1}^{∞} ln(1 − x^{i}),

ln(H(x)) = ln 49 + ln x + 7 ∑_{i=1}^{∞} ln(1 − x^{7i}) − 8 ∑_{i=1}^{∞} ln(1 − x^{i}).

By using the power series expansion of ln(1 − x), we get

ln(G(x)) = ln 7 − 3 ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{x^{7ij}}{j} + 4 ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{x^{ij}}{j} ,  (10)

ln(H(x)) = ln 49 + ln x − 7 ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{x^{7ij}}{j} + 8 ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{x^{ij}}{j} .  (11)

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

f_{1}(x) = ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{x^{7ij}}{j} ,  (12)

f_{2}(x) = ∑_{i=1}^{∞} ∑_{j=1}^{∞} \frac{x^{ij}}{j} .  (13)

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

f_{1}(x) = ∑_{i=1}^{∞} a_{i}x^{i}.

Then,

a_{i} = \begin{cases}\frac{σ(i/7)}{i/7}, & if 7|i; \\ 0, & otherwise. \end{cases}  (14)

We write the function f_{2}(x) as

f_{2}(x) = ∑_{i=1}^{∞} b_{i}x^{i}

where

b_{i} = \frac{σ(i)}{i}.  (15)

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

ln(G(x)) = ln 7 + ∑_{i=1}^{∞} d_{i}x^{i}  (16)

where

d_{i} = \begin{cases}\frac{4σ(i)}{i} − \frac{3σ(i/7)}{i/7}, & if 7|i; \\ \frac{4σ(i)}{i}, & otherwise. \end{cases}  (17)

By Lemma 1, we write (17) as

d_{i} = \frac{σ(i)}{i}(1 + \frac{18}{7^{m+1} − 1})  (18)

for i = 7^{m}i′, m ≥ 0. Similarly, by (11), (12), (13), (14), (15), we get

ln(H(x)) = ln 49 + ln x + ∑_{i=1}^{∞} e_{i}x^{i}  (19)

where

e_{i} = \begin{cases}\frac{8σ(i)}{i} − \frac{7σ(i/7)}{i/7}, & if 7|i; \\ \frac{8σ(i)}{i}, & otherwise. \end{cases}  (20)

By Lemma 1, we write (20) as

e_{i} = \frac{σ(i)}{i}(1 + \frac{42}{7^{m+1} − 1})  (21)

for i = 7^{m}i′, m ≥ 0. By (1), (16), (18), we have

G(x) = exp(ln G(x)) = 7 exp ( ∑_{n=1}^{∞} d_{n}x^{n}) = 7 exp ( ∑_{n=1}^{∞}(n!d_{n})\frac{x^{n}}{n!} )

= 7( ∑_{n=0}^{∞} B_{n}(1!d_{1}, 2!d_{2}, …, n!d_{n})\frac{x^{n}}{n!} ).  (22)

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

H(x) = exp(ln H(x)) = 49x exp ( ∑_{n=1}^{∞} e_{n}x^{n}) = 49x exp ( ∑_{n=1}^{∞}(n!e_{n})\frac{x^{n}}{n!} )

= 49x( ∑_{n=0}^{∞} B_{n}(1!e_{1}, 2!e_{2}, …, n!e_{n})\frac{x^{n}}{n!} )

= 49( ∑_{n=1}^{∞} B_{n−1}(1!e_{1}, 2!e_{2}, …, (n − 1)!e_{n−1})\frac{x^{n}}{(n − 1)!}).  (23)

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

∑_{n=1}^{∞} p(7n + 5)x^{n} = ∑_{n=1}^{∞} (7B_{n}(1!d_{1}, 2!d_{2}, …, n!d_{n}) + 49nB_{n−1}(1!e_{1}, 2!e_{2}, …, (n − 1)!e_{n−1}))\frac{x^{n}}{n!}

as desired.

3 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.

References

[1] G. Andrews, The Theory of Partitions, Cambridge University Press, 1976.

[2] E. T. Bell, Exponential polynomials, Ann. Math. 35 (1934), 258–277.

[3] S. Bouroubi and N. Benyahia Tani, A new identity for complete Bell polynomials based on a formula of Ramanujan, J. Integer Sequences 12 (2009), Article 09.3.5.

[4] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., 1974.

[5] S. Ramanujan, Congruence properties of partitions, Math. Z. 9 (1921), 147–153.

[6] J. Riordan, Combinatorial Identities, Robert E. Krieger Publishing Co., 1979.

2010 Mathematics Subject Classification: Primary 05A17; Secondary 11P81. Keywords: Bell polynomial, integer partition, divisor sum, Ramanujan’s congruence.

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

Received February 24 2018; revised versions received June 2 2018; June 3 2018. Published in Journal of Integer Sequences, August 22 2018.

Return to Journal of Integer Sequences home page.