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

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.

The Numerical Evaluation Methods for Beta Function

“…It is used in applied and engineering mathematics, statistics and probability, and computational physics. Recently, there is a great interest in the mathematical applications of beta function, some of these studies: [2][3][4][5][6][7][8][9]. Besides, in physics, the researchers observed that many properties of the strong nuclear force are defined by beta function, based on the data which are obtained during their research at CERN (The European Organization for Nuclear Research).…”