Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions

Feng Qi; Chuan-Jun Huang

doi:10.1007/s13398-020-00927-y

Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions

Rev R Acad Cienc Exactas Fis Nat A Mat. 2020;114(4):191. doi: 10.1007/s13398-020-00927-y. Epub 2020 Aug 24.

Authors

Feng Qi^{1

2

3}, Chuan-Jun Huang⁴

Affiliations

¹ Institute of Mathematics, Henan Polytechnic University, Jiaozuo, 454010 Henan China.
² College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao, 028043 Inner Mongolia China.
³ School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin, 300387 China.
⁴ Department of Mathematics, Ganzhou Teachers College, Ganzhou, 341000 Jiangxi China.

Abstract

In the paper, by virtue of the binomial inversion formula, a general formula of higher order derivatives for a ratio of two differentiable function, and other techniques, the authors compute several sums in terms of the beta function and its partial derivatives, polygamma functions, the Gauss hypergeometric function, and a determinant. These results generalize known ones in combinatorics.

Keywords: Beta function; Binomial inversion formula; Derivative formula for a ratio of two differential functions; Determinant; Gauss hypergeometric function; Identity; Polygamma function; Sum.