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Let the numbers be defined by <br/> , where <br/> 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.

Let

In [

where

Spiess [

where

The paper is organized as follows. In Section 2, we obtain some for

involving the numbers

Hence we write

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

Lemma 1 (see [

Theorem 1. Let

Proof. By (1), we have

Comparing the coefficients of

Recall that

Corollary 1. Let

Theorem 2. Let

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

Theorem 3. Let

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

which is just the desired result.

Setting

Corollary 2 Let

Corollary 3 Let

Proof. Setting

Corollary 4. Let

Proof. Setting

Theorem 4. Let

Proof. which is just the desired result.

Setting

Corollary 5. Let

Corollary 6. The substitutions

Setting

Corollary 7. Let

Theorem 5. Let

where

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

which is just the desired result.

Setting

Corollary 8. Let

Setting

Corollary 9. Let

We give four applications of Corollary 9:

Corollary 10. Let

For identities involving Harmonic numbers and inverse of binomial coefficients

In Section, we obtain some for

In [

From the generating function of

Theorem 6. For

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

This gives (11).

Corollary 11 Setting

Setting

Corollary 12 The following relation holds

Corollary 13. The following relation holds

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 [

The next, we gives identities related to

For completeness we supply proofs:

Similarly, we obtain summation formulas related

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

Similarly, for completeness we supply a proof:

By (28) minus (30), we get

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

Theorem 7 For

Proof. By Lemma 1, we have

and this complete the proof.

Similarly, we can obtain the next Theorem.

Theorem 8. Let

Theorem 9. For

Proof. By Lemma 1, we have

this give (38).

Theorem 10. For

Proof. By Corollary 3 of [

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.