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


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KÜÇÜKOĞLU İ., Simsek B., ŞİMŞEK Y.

TURKISH JOURNAL OF MATHEMATICS, vol.43, no.5, pp.2337-2353, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 43 Issue: 5
  • Publication Date: 2019
  • Doi Number: 10.3906/mat-1906-6
  • Journal Name: TURKISH JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, TR DİZİN (ULAKBİM)
  • Page Numbers: pp.2337-2353
  • Keywords: Generating functions, Stirling numbers, Apostol-Bernoulli numbers, Apostol-Euler numbers, Catalan numbers, combinatorial sums, binomial coefficients, Chu-Vandermonde convolution formula, probability distribution, APOSTOL-TYPE NUMBERS, EULER POLYNOMIALS, BERNOULLI
  • Akdeniz University Affiliated: Yes

Abstract

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.