American Journal of Computational Mathematics
Vol.05 No.02(2015), Article ID:56615,9 pages
10.4236/ajcm.2015.52008
Sums of Involving the Harmonic Numbers and the Binomial Coefficients
Wuyungaowa, Sudan Wang
School of Mathematical Sciences, Inner Mongolia University, Hohhot, China
Email: wuyungw@163.com
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
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
- Zave, D.A. (1976) A Series Expansion Involving the Harmonic Numbers. Information Processing Letters, 5, 75-77. http://dx.doi.org/10.1016/0020-0190(76)90068-5
- Spiess, J. (1990) Some Identities Involving Harmonic Numbers. Mathematics Computation, 55, 839-863. http://dx.doi.org/10.1090/S0025-5718-1990-1023769-6
- Brietzke, E.H.M. (2008) An Identity of Andrews and a New Method for the Riordan Array Proof of Combinatorial Identities. Discrete Mathematics, 308, 4246-4262. http://dx.doi.org/10.1016/j.disc.2007.08.050
- Wang, W. and Wang, T (2008) Generalized Riordan Arrays. Discrete Mathematics, 308, 6466-6500. http://dx.doi.org/10.1016/j.disc.2007.12.037
- Flajolet, P., Fusy, E., Gourdon, X., Panario, D. and Pouyanne, N. (2006) A Hybrid of Darboux’s Method and Singularity Analysis in Combinatorial Asymptotics. The Electronic Journal of Combinatorics, 13.
- Sofo, A. (2012) Euler Related Sums. Mathematical Sciences, 6, 10.
- Sury, B. (1993) Sum of the Reciprocals of the Binomial Coefficients. European Journal of Combinatorics, 14, 351- 353. http://dx.doi.org/10.1006/eujc.1993.1038
- Jonathan, M. (2009) Borwein and O-Yeat Chang. Duallity in Tails of Multiple-Zeta Values, 54, 2220-2234.
- David, B. and Borwein, J.M. (1995) On an Intrguing Integral and Some Series Relate to. Proceedings of the American Mathematical Society, 123, 1191-1198.>http://html.scirp.org/file/5-1100402x159.png" class="200" />. Proceedings of the American Mathematical Society, 123, 1191-1198.
- Flajolet, P. and Sedgewick, R. (1995) Mellin Transforms Asymptotics: Finite Differences and Rice’s Integrals. Theoretical Computer Science, 144, 101-124. http://dx.doi.org/10.1016/0304-3975(94)00281-M