**Title:** A SIMPLE PROOF OF A BINOMIAL IDENTITY WITH APPLICATIONS

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-19-00008; **Volume 1 / Issue 2 / Year 2019**, Pages 13-20

**Document Type:** Research Paper

**Author(s):** Ting-Ting Bai ^{a} , Qiu-Ming Luo ^{b}

^{a}Department of Mathematics, Chongqing Normal University Chongqing Higher Education Mega Center, Huxi Campus Chongqing 401331, People’s Republic of China

^{b}Department of Mathematics, Chongqing Normal University Chongqing Higher Education Mega Center, Huxi Campus Chongqing 401331, People’s Republic of China

Received: 5 December 2019, Accepted: 12 December 2019, Available online: 28 December 2019.

**Corresponding Author:** Qiu-Ming Luo (Email address: luomath2007@163.com)

**Full Text:** PDF

**Abstract**

Peterson [Amer. Math. Monthly, 120 (2013), 558–562] gave a probabilistic proof of a binomial identity. In this paper, by using the partial fraction decomposition, we give a simple proof of this binomial identity. As some applications, we obtain some interesting harmonic number identities.

**Keywords:** Binomial identity, Harmonic number, Bell polynomial, Partial fraction decomposition

**References:**

- J. Choi,
*Certain summation formulas involving harmonic numbers and generalized harmonic numbers*, Applied Mathematics and Computation**218**, 734-740, 2011. - J. Choi, H. M. Srivastava,
*Some summation formulas involving harmonic numbers and generalized harmonic numbers*, Mathematical and Computer Modelling**54**, 2220-223, 2011. - L. Comtet,
*Advanced Combinatorics: The Art of Finite and Infinite Expansions*, Springer, Reidel, Dordrecht and Boston, 1974. - H.W. Gould,
*Combinatorial Identities: Astandardized Set of Tables Listing 500 Binomial Coefficient Summations*, Morgantown, W. Va. 1972. - J. Peterson,
*A probabilistic proof of a binomial identity*, Amer. Math. Monthly**120**, 558-562, 2013. - T. M. Rassias, H. M. Srivastava,
*Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers*, Applied Mathematics and Computation**131**, 593-605, 2002.