A New Identity for Complete Bell Polynomials Based on a Formula of

Ramanujan

Sadek Bouroubi

University of Science and Technology Houari Boumediene

Faculty of Mathematics

Laboratory LAID3

P. O. Box 32

16111 El-Alia, Bab-Ezzouar, Algiers

Algeria

sbouroubi@usthb.dz
bouroubis@yahoo.fr

Nesrine Benyahia Tani

Faculty of Economics and Management Sciences Laboratory LAID3 Ahmed Waked Street Dely Brahim, Algiers

Algeria

benyahiatani@yahoo.fr

Abstract

Let p(n) be the number of partitions of n. In this paper, we give a new identity for complete Bell polynomials based on a sequence related to the generating function of p(5n + 4) established by Srinivasa Ramanujan.

1 Introduction

Let us first present some necessary definitions related to the Bell polynomials, which are quite general and have numerous applications in combinatorics. For a more complete exposition, the reader is referred to the excellent books of Comtet [4], Riordan [6] and Stanley [9].

Let (a_{1}, a_{2}, …) be a sequence of real or complex numbers. Its partial (exponential) Bell polynomial B_{n,k}(a_{1}, a_{2}, …), is defined as follows:

∑^{∞} ; n=kB_{n,k} (a_{1}, a_{2}, …) \frac{t^{n}}{n!} = \frac{1}{k!} (∑^{∞} ; m=1a_{m} \frac{t^{m}}{m!})^{k} ·

Their exact expression is

B_{n,k} (a_{1}, a_{2}, …) = ∑_{π(n,k)} \frac{n!}{k_{1}!k_{2}! ⋯} (\frac{a_{1}}{1!} )^{k_{1}} (\frac{a_{2}}{2!} )^{k_{2}} ⋯,

where π (n, k) denotes the set of all integer solutions (k_{1}, k_{2}, …) of the system

\{ \begin{matrix}k_{1} + ⋯ + k_{j} + ⋯ = k; \\ k_{1} + ⋯ + jk_{j} + ⋯ = n. \end{matrix}

The (exponential) complete Bell polynomials are given by

exp ( ∞ ; ∑_{m=1}a_{m} \frac{t^{m}}{m!}) = ∞ ; ∑_{n=0}A_{n}(a_{1}, a_{2}, …)\frac{t^{n}}{n!}·

In other words,

A_{0}(a_{1}, a_{2}, …) = 1 and A_{n}(a_{1}, a_{2}, …) = ∑_{k=1}^{n} B_{n,k}(a_{1}, a_{2}, …), ∀n ≥ 1.

Hence

A_{n} (a_{1}, a_{2}, …) = ∑_{k_{1}+...+jk_{j}+...=n} \frac{n!}{k_{1}!k_{2}! ⋯} (\frac{a_{1}}{1!} )^{k_{1}} (\frac{a_{2}}{2!} )^{k_{2}} ⋯

The main tool used to prove our main result in the next section is the following formula of Ramanujan, about which G. H. Hardy [5] said: “... but here Ramanujan must take second place to Prof. Rogers; and if I had to select one formula from all of Ramanujan’s work, I would agree with Major MacMahon in selecting ...

∑_{n=0}^{∞} p(5n + 4) x^{n} = \frac{5\{(1 − x^{5})(1 − x^{10})(1 − x^{15}) ⋯ \}^{5}}{\{(1 − x)(1 − x^{2})(1 − x^{3}) ⋯ \}^{6}} ,  (1)

where p(n) is the number of partitions of n.”

2 Some basic properties of the divisor function

Let σ(n) be the sum of the positive divisors of n. It is clear that σ(p) = 1 + p for any prime number p, since the only positive divisors of p are 1 and p. Also the only divisors of p^{2} are 1, p and p^{2}. Thus

σ(p^{2}) = 1 + p + p^{2} = \frac{p^{3} − 1}{p − 1}·

It is now easy to prove [8]

σ(p^{k}) = \frac{p^{k+1} − 1}{p − 1}·  (2)

It is well known in number theory [8] that σ(n) is a multiplicative function, that is, if n and m are relatively prime, then

σ(nm) = σ(n) σ(m).  (3)

An immediate consequence of these facts is the following Lemma:

Lemma 1. If 5 | n, then it exists α ≥ 1, so that

σ(n) = \frac{5^{α+1} − 1}{5^{α} − 1} σ (\frac{n}{5}) .  (4)

where α is the power to which 5 occur in the decomposition of n into prime factors.

Proof. From (2) and (3), we have

\begin{aligned}σ(n) & = & \frac{5^{α+1} − 1}{4} σ (\frac{n}{5^{α}} ) , and \\ σ (\frac{n}{5}) & = & \frac{5^{α} − 1}{4} σ (\frac{n}{5^{α}} ) . \end{aligned}

Hence the result follows.

3 Main result

Henceforth, let us express n by n = 5^{α} · p_{1}^{α1} · p_{2}^{α2} ⋯ p_{r}^{α_{r}}, where the p′s are distinct primes different from 5, and the α′s are the powers to which they occur.

The present theorem is the main result of this work.

Theorem 2. Let a(n) be the real number defined as follows:

a_{n} = (1 + \frac{20}{5^{α+1} − 1}) \frac{σ(n)}{n}·

Then we have

A_{n}(1!a_{1}, 2!a_{2}, …, n!a_{n}) = \frac{n!}{5} p(5n + 4)·

Proof. Put

g(x) = \frac{5\{(1 − x^{5})(1 − x^{10})(1 − x^{15}) ⋯ \}^{5}}{\{(1 − x)(1 − x^{2})(1 − x^{3}) ⋯ \}^{6}} ·

Then

ln(g(x)) = ln 5 + 5 ln ∏_{i=1}^{∞}(1 − x^{5i}) − 6 ln ∏_{i=1}^{∞}(1 − x^{i})
= ln 5 + 5 ∑_{i=1}^{∞} ln(1 − x^{5i}) − 6 ∑_{i=1}^{∞} ln(1 − x^{i})
= ln 5 − 5 ∑_{i,j=1}^{∞} \frac{x^{5ij}}{j} + 6 ∑_{i,j=1}^{∞} \frac{x^{ij}}{j}
= ln 5 + ∑_{n=1}^{∞} a_{n}x^{n},

(5)

where

a_{n} = \{ \begin{matrix}\frac{6 σ(n) − 25 σ (\frac{n}{5})}{n}, & if 5 | n; \\ \frac{6}{n} σ(n), & otherwise. \end{matrix}

If α ≥ 1, i.e., 5 | n, then we get by (4)

σ (\frac{n}{5}) = \frac{5^{α} − 1}{5^{α+1} − 1} σ(n).

Thus

a_{n} = (1 + \frac{20}{5^{α+1} − 1}) \frac{σ(n)}{n}, ∀α ≥ 0.

Hence, we obtain from (5)

g(x) = 5. exp ( ∑_{n=1}^{∞} a_{n}x^{n})
= 5 (1 + ∑_{k=1}^{∞} \frac{(∑_{n=1}^{∞} n!a_{n} \frac{x^{n}}{n!} )^{k}}{k!} )
= 5 + 5 ∑_{k=1}^{∞} (∑_{n=k}^{∞} B_{n,k}(1!a_{1}, 2!a_{2}, …)\frac{x^{n}}{n!} )
= 5 + 5 ∑_{n=1}^{∞} A_{n}(1!a_{1}, 2!a_{2}, …, n!a_{n})\frac{x^{n}}{n!}
= 5 ∑_{n=0}^{∞} A_{n}(1!a_{1}, …, n!a_{n})\frac{x^{n}}{n!} ·

Therefore, by comparing coefficients of the two power series in (1), we finally get

A_{n}(1!a_{1}, 2!a_{2}, …, n!a_{n}) = \frac{n!}{5} p(5n + 4), for n ≥ 0.

Theorem 2 has the following Corollary. Corollary 3. For n ≥ 1, we have

σ(n) = \frac{5^{α+1} − 1}{5^{α+1} + 19} · \frac{1}{(n − 1)!} ∑_{j=1}^{n}(−1)^{j−1}(j − 1)! B_{n,j} (\frac{1!}{5}p(9), \frac{2!}{5}p(14), …) .

Proof. This follows from the following inversion relation of Chaou and al [3]:

y_{n} = ∑_{k=1}^{n} B_{n,k}(x_{1}, x_{2}, …) ⇔ x_{n} = ∑_{k=1}^{n}(−1)^{k−1}(k − 1)! B_{n,k}(y_{1}, y_{2}, …).

4 Acknowledgements

The authors would like to thank the referee for his valuable comments which have improved the quality of the paper.

References

[1] M. Abbas and S. Bouroubi, On new identities for Bell’s polynomials, Discrete Mathematics 293 (2005) 5–10.

[2] E. T. Bell, Exponential polynomials, Annals of Mathematics 35 (1934), 258–277.

[3] W.-S. Chaou, Leetsch C. Hsu, and Peter J.-S. Shiue, Application of Faà di Bruno’s formula in characterization of inverse relations. Journal of Computational and Applied Mathematics 190 (2006), 151–169.

[4] L. Comtet, Advanced Combinatorics. D. Reidel Publishing Company, Dordrecht-Holland, Boston, 1974, pp. 133–175.

[5] G. H. Hardy, Ramanujan, Amer. Math. Soc., Providence, 1999.

[6] J. Riordan, Combinatorial Identities, Huntington, New York, 1979.

[7] J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, New York, 1958; Princeton University Press, Princeton, NJ, 1980.

[8] K. Rosen, Elementary Number Theory and Its Applications, 4th ed., Addison-Wesley, 2000.

[9] R. P. Stanley, Enumerative Combinatorics, Volume 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, Cambridge, 1997.

2000 Mathematics Subject Classification: 05A16, 05A17, 11P81 Keywords: Bell polynomials, integer partition, Ramanujan’s formula.

(Concerned with sequences \underline{A000041} and \underline{A071734}.)

Received January 3 2009; revised version received March 22 2009. Published in Journal of Integer Sequences March 27 2009.

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