The aim of this paper is to not only provide a definition of a new family of special numbers and polynomials of higher-order with their generating functions, but also to investigate their fundamental properties in the spirit of probabilistic distributions. By applying generating functions methods, we derive miscellaneous novel identities and formulas 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 which returns special values of these numbers and polynomials. In addition, we show that our new identities and formulas are connected with the interpolation functions of the Apostol-type numbers and polynomials. Finally, we present some theoretical and applied details on probabilistic distributions arising from the aforementioned Chu-Vandermonde-type convolution formulas.
Generating functions, Stirling numbers, Apostol-Bernoulli numbers, Apostol-Euler numbers, Catalan numbers, combinatorial sums, binomial coefficients, Chu-Vandermonde convolution formula, probability distribution
KÜÇÜKOĞLU, İREM; ŞİMŞEK, BURÇİN; and ŞİMŞEK, YILMAZ
"An approach to negative hypergeometric distribution by generatingfunction for special numbers and polynomials,"
Turkish Journal of Mathematics: Vol. 43:
5, Article 21.
Available at: https://journals.tubitak.gov.tr/math/vol43/iss5/21