Received 5 December 2014; accepted 22 May 2015; published 25 May 2015

ABSTRACT

Let the numbers be defined by, where and are the exponential complete Bell polynomials. In this paper, by means of the methods of Riordan arrays, we establish general identities involving the numbers, binomial coefficients and inverse of binomial coefficients. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and the Euler sum identities. Furthermore, we obtain the asymptotic values of some summations associated with the numbers by Darboux’s method.

Keywords:

Harmonic Numbers, Euler Sum, Riordan Arrays, Asymptotic Values

1. Introduction and Preliminaries

Let be the exponential complete Bell polynomials and

In [1] , Zave established the following series expansion:

(1)

where for, and.

Spiess [2] introduced the numbers and, for; then Equation (1.1) is equivalent to

where, , ,

The paper is organized as follows. In Section 2, we obtain some for and binomial coefficients by means of the Riordan arrays. In Section 3, we establish some identities involving the numbers and inverse of binomial coefficients. Finally, in Section 4, we give the asymptotic expansions of some summations

involving the numbers by Darboux’s method. Due to [3] [4] , a Riordan array is a pair of formal power series with. It defines an infinite lower triangular array according to the rule

Hence we write. If is an Riordan array and is the generating function of the sequence, i.e.,. Then we have

(2)

Based on the generating function (1), we obtain the next Riordan arrays, to which we pay particular attention in the present paper:

(3)

Lemma 1 (see [5] ) Let be a real number and. When,

2. Identities Involving the Numbers and Binomial Coefficients

Theorem 1. Let, , , then

(4)

Proof. By (1), we have

(5)

Comparing the coefficients of on both sides of (5), we completes the proof of Theorem 1.

Recall that Thus, setting in Theorem 1 gives the next three identities, respectively.

Corollary 1. Let, , the following relations hold

Theorem 2. Let, , then

(6)

Proof. To obtain the result, make use of the Theorem 1.

Theorem 3. Let, , then

(7)

Proof. Applying the summation property (2) to the Riordan arrays (3), we have

which is just the desired result.

Setting in Theorem 3 gives the next Corollary.

Corollary 2 Let, then

Corollary 3 Let, , then

Proof. Setting in Theorem 3 gives Corollary 3.

Corollary 4. Let, then

Proof. Setting in Corollary 2 yields Corollary 4.

Theorem 4. Let, , then

(8)

Proof. which is just the desired result.

Setting in Theorem 4 gives the next Corollary.

Corollary 5. Let, then

Corollary 6. The substitutions in Theorem 4 gives the next four identities, respectively.

Setting in Corollary 5 gives the next four identities, respectively.

Corollary 7. Let, then

Theorem 5. Let, , then

(9)

where are the Stirling numbers of the first kind.

Proof. By (1) and (2), we have

which is just the desired result.

Setting in Theorem 5 gives the next Corollary.

Corollary 8. Let, then

Setting in Theorem 6 gives the next Corollary.

Corollary 9. Let, , then

We give four applications of Corollary 9:

Corollary 10. Let, then

3. Identities Involving and Inverse of Binomial Coefficients

For identities involving Harmonic numbers and inverse of binomial coefficients in given in [6] .

In Section, we obtain some for and binomial coefficients by means of the Riordan arrays. From these identities, we deduce some identities involving binomial coefficients, Harmonic numbers and identities related to,

In [7] , the inverse of a binomial coefficient is related to an integral, as follows

(10)

From the generating function of and (10), we have

Theorem 6. For be any integer, then

(11)

Proof. From (1) and (10), we obtain

This gives (11).

Corollary 11 Setting in Theorem 6, The following relation holds:

(12)

(13)

(14)

(15)

Setting in Corollary 11, gives the next identities.

Corollary 12 The following relation holds

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

Corollary 13. The following relation holds

(24)

(25)

(26)

(27)

Proof. (16) minus(20) give (24); (17) minus (21), (18) minus (22) and (19) minus (23), yields (25), (26) and (27), respectively.

Leonhard Euler (1707-1783) had already stated the equation

Recall the Euler sum identities [8] [9] .

The next, we gives identities related to,

For completeness we supply proofs:

(28)

(29)

Similarly, we obtain summation formulas related,

(30)

(31)

By (18) and (28), (19) and (31), we have

(32)

(33)

Similarly, for completeness we supply a proof:

(34)

By (28) minus (30), we get

(35)

Applying (25) and (34), (26) and (32), we have

4. Asymptotics

Theorem 7 For be any integer, as, we have

(36)

Proof. By Lemma 1, we have

and this complete the proof.

Similarly, we can obtain the next Theorem.

Theorem 8. Let be any integer, as, we have

(37)

Theorem 9. For be any integer, as, we have

(38)

Proof. By Lemma 1, we have

this give (38).

Theorem 10. For be any integer, as, we have

(39)

Proof. By Corollary 3 of [10] , immediately complete the proof of Theorem 10.

Acknowledgements

The author would like to thank an anonymous referee whose helpful suggestions and comments have led to much improvement of the paper. The research is supported by the Natural Science Foundation of China under Grant 11461050 and Natural Science Foundation of Inner Mongolia under Grant 2012MS0118.

References