An Explicit Formula for Bell Numbers in Terms of

An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions Feng Qi Department of Mathematics, College of Science Tianjin Polytechnic University Tianjin City, 300387, People’s Republic of China [email protected] [email protected] [email protected] http://qifeng618.wordpress.com Abstract In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.

1

Introduction

In combinatorics, Bell numbers, usually denoted by Bn for n ∈ {0} ∪ N, count the number of ways a set with n elements can be partitioned into disjoint and non-empty subsets. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. Every Bell number Bn may be generated by ee

x −1

=

∞ X k=0

or, equivalently, by e

e−x −1

=

∞ X

Bk

xk k!

(−1)k Bk

k=0

(1)

xk . k!

(2)

In combinatorics, Stirling numbers arise in a variety of combinatorics problems. They are introduced in the eighteen century by James Stirling. There are two kinds of Stirling 1

numbers: Stirling numbers of the first kind and Stirling numbers of the second kind. Every Stirling number of the second kind, usually denoted by S(n, k), is the number of ways of partitioning a set of n elements into k nonempty subsets, may be computed by   k 1 X i k (k − i)n , (−1) S(n, k) = i k! i=0

(3)

and may be generated by ∞

(ex − 1)k X xn = S(n, k) , k! n! n=k

k ∈ {0} ∪ N.

(4)

In the theory of special functions, the hypergeometric functions are denoted and defined by p Fq (a1 , . . . , ap ; b1 , . . . , bq ; x)

=

∞ X (a1 )n · · · (ap )n xn n=0

(b1 )n · · · (bq )n n!

(5)

for bi ∈ / {0, −1, −2, . . . } and p, q ∈ N, where (a)0 = 1 and (a)n = a(a + 1) · · · (a + n − 1) for n ∈ N and any complex number a is called the rising factorial. Specially, the series 1 F1 (a; b; z) =

∞ X (a)k z k k=0

(b)k k!

(6)

is called Kummer confluent hypergeometric function. In combinatorics or number theory, it is common knowledge that Bell numbers Bn may be computed in terms of Stirling numbers of the second kind S(n, k) by Bn =

n X

S(n, k).

(7)

k=1

In this paper, we will find a new explicit formula for computing Bell numbers Bn in terms of Kummer confluent hypergeometric functions 1 F1 (k + 1; 2; 1) and Stirling numbers of the second kind S(n, k) as follows. Theorem 1. For n ∈ N, Bell numbers Bn may be expressed as n

1X Bn = (−1)n−k S(n, k)k!1 F1 (k + 1; 2; 1). e k=1

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Proof of Theorem 1

We now start out to verify Theorem 1 as follows.

2

(8)

Among other things, Qi and Wang [6, Theorem 1.2] obtained that the function Hk (z) = e

k X 1 1 − m! z m m=0

1/z

for k ∈ {0} ∪ N and z 6= 0 has the integral representation Z ∞ 1 k −zt d t, Hk (z) = 1 F2 (1; k + 1, k + 2; t)t e k!(k + 1)! 0

(9)

0.

(10)

See also [4, Section 1.2] and [5, Lemma 2.1]. When k = 0, the integral representation (10) becomes √
Z ∞ I1 2 t −zt 1/z √ e =1+ e d t, 0, (11) t 0 where Iν (z) stands for the modified Bessel function of the first kind Iν (z) =

∞ X k=0

 2k+ν z 1 k!Γ(ν + k + 1) 2

(12)

for ν ∈ R and z ∈ C, see [1, p. 375, 9.6.10], and Γ represents the classical Euler gamma function which may be defined by Z ∞ tz−1 e−t d t, 0, (13) Γ(z) = 0

see [1, p. 255]. Replacing z by ex in (11) gives 1/ex

e

=e

e−x

∞

Z =1+ 0

√
I1 2 t −ex t √ e d t. t

Differentiating n ≥ 1 times with respect to x on both sides of (14) and (2) gives √
Z ∞ −x x I1 2 t dn e−e t dn ee √ = dt d xn d xn t 0 and

−x ∞ X dn ee xk−n k = e (−1) B . k d xn (k − n)! k=n

From (15) and (16), it follows that ∞ X

xk−n e (−1)k Bk = (k − n)! k=n

Z 0

3

∞

√
x I1 2 t dn e−e t √ d t. d xn t

(14)

(15)

(16)

Taking x → 0 in the above equation yields ∞

Z

n

(−1) eBn = 0

√
x I1 2 t dn e−e t √ lim d t. x→0 d xn t

(17)

In combinatorics, Bell polynomials of the second kind, or say, the partial Bell polynomials, Bn,k (x1 , x2 , . . . , xn−k+1 ) are defined by n−k+1 Y 

n!

X

Bn,k (x1 , x2 , . . . , xn−k+1 ) =

1≤i≤n,` i ∈N Pn i`i =n Pi=1 n i=1 `i =k

Qn−k+1 i=1

`i !

i=1

xi `i i!

for n ≥ k ≥ 1, see [2, p. 134, Theorem A], and satisfy
Bn,k abx1 , ab2 x2 , . . . , abn−k+1 xn−k+1 = ak bn Bn,k (x1 , xn , . . . , xn−k+1 ) and

(18)

(19)

n−k+1

z }| { Bn,k (1, 1, . . . , 1) = S(n, k),

(20)

see [2, p. 135], where a and b are any complex numbers. The well-known Fa`a di Bruno formula may be described in terms of Bell polynomials of the second kind Bn,k (x1 , x2 , . . . , xn−k+1 ) by n X
dn f ◦ g(x) = f (k) (g(x))Bn,k g 0 (x), g 00 (x), . . . , g (n−k+1) (x) , n dx k=1

(21)

see [2, p. 139, Theorem C]. By Fa`a di Bruno formula (21) and the identities (19) and (20), we have n−k+1 x n z }| { dn e−e t X −ex t x x x e B ( = −e t, −e t, . . . , −e t) n,k d xn k=1 n−k+1

−ex t

=e

n X z }| { (−ex t)k Bn,k (1, 1, . . . , 1)

(22)

k=1

= e−e

xt

n X

(−ex t)k S(n, k)

k=1 n X −t →e (−t)k S(n, k) k=1

as x → 0. Substituting (22) into (17) leads to n

1X Bn = (−1)n−k S(n, k) e k=1 4

Z 0

∞

√
I1 2 t tk−1/2 e−t d t

n

=

1X (−1)n−k S(n, k)k!1 F1 (k + 1; 2; 1). e k=1

The required proof is complete. Remark 1. This paper is a slightly modified version of the preprint [3].

References [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 10th printing, Dover Publications, 1972. 3 [2] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. 4 [3] F. Qi, An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions, available online at http://arxiv.org/abs/1402.2361. 5 [4] F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, preprint, http://arxiv.org/abs/ 1302.6731. 3 [5] F. Qi and C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math. 10 (2013), 1685–1696; Available online at http://dx.doi.org/10.1007/ s00009-013-0272-2. 3 [6] F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, preprint, http://arxiv.org/abs/1210.2012. 3

2010 Mathematics Subject Classification: Primary 11B73; Secondary 33B10, 33C15. Keywords: explicit formula, Bell number, confluent hypergeometric function of the first kind, Stirling number of the second kind. (Concerned with sequences A000110 and A008277.)

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