An approach to negative hypergeometric distribution by generating function for special numbers and polynomials

Abstract

The aim of this paper is to not only provide a definition of a new family of special numbers and polynomialsof higher-order with their generating functions, but also to investigate their fundamental properties in the spirit ofprobabilistic distributions. By applying generating functions methods, we derive miscellaneous novel identities andformulas involving the Chu–Vandermonde-type convolution formulas, combinatorial sums, Bernstein basis functions,and the other well-known special numbers and polynomials. Moreover, we provide a computational algorithm whichreturns special values of these numbers and polynomials. In addition, we show that our new identities and formulasare connected with the interpolation functions of the Apostol-type numbers and polynomials. Finally, we present sometheoretical and applied details on probabilistic distributions arising from the aforementioned Chu–Vandermonde-typeconvolution formulas.