Sums of Involving the Harmonic Numbers and the Binomial Coefficients ()
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.