AJCM  Vol.5 No.2 , June 2015
Sums of Involving the Harmonic Numbers and the Binomial Coefficients
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.
Cite this paper: Wuyungaowa, &. and Wang, S. (2015) Sums of Involving the Harmonic Numbers and the Binomial Coefficients. American Journal of Computational Mathematics, 5, 96-105. doi: 10.4236/ajcm.2015.52008.

[1]   Zave, D.A. (1976) A Series Expansion Involving the Harmonic Numbers. Information Processing Letters, 5, 75-77.

[2]   Spiess, J. (1990) Some Identities Involving Harmonic Numbers. Mathematics Computation, 55, 839-863.

[3]   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.

[4]   Wang, W. and Wang, T (2008) Generalized Riordan Arrays. Discrete Mathematics, 308, 6466-6500.

[5]   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.

[6]   Sofo, A. (2012) Euler Related Sums. Mathematical Sciences, 6, 10.

[7]   Sury, B. (1993) Sum of the Reciprocals of the Binomial Coefficients. European Journal of Combinatorics, 14, 351- 353.

[8]   Jonathan, M. (2009) Borwein and O-Yeat Chang. Duallity in Tails of Multiple-Zeta Values, 54, 2220-2234.

[9]   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.

[10]   Flajolet, P. and Sedgewick, R. (1995) Mellin Transforms Asymptotics: Finite Differences and Rice’s Integrals. Theoretical Computer Science, 144, 101-124.