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Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function

Kucukoglu, Irem and Simsek, Burcin and Simsek, Yilmaz (2019) Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function. Axioms, 8 (4). p. 112. ISSN 2075-1680

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Abstract

The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson&#8722;Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the <i>p</i>-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.

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Details

Item Type: Article
Status: Published
Creators/Authors:
CreatorsEmailPitt UsernameORCID
Kucukoglu, Irem
Simsek, Burcin
Simsek, Yilmaz
Date: 11 October 2019
Date Type: Publication
Journal or Publication Title: Axioms
Volume: 8
Number: 4
Publisher: MDPI AG
Page Range: p. 112
DOI or Unique Handle: 10.3390/axioms8040112
Schools and Programs: Dietrich School of Arts and Sciences > Statistics
Refereed: Yes
Uncontrolled Keywords: generating functions, functional equations, partial differential equations, special numbers and polynomials, bernoulli numbers, euler numbers, stirling numbers, bell polynomials, cauchy numbers, poisson-charlier polynomials, bernstein basis functions, daehee numbers and polynomials, combinatorial sums, binomial coefficients, p-adic integral, probability distribution
ISSN: 2075-1680
Official URL: http://dx.doi.org/10.3390/axioms8040112
Article Type: Research Article
Date Deposited: 25 May 2021 19:37
Last Modified: 25 May 2021 19:37
URI: http://d-scholarship.pitt.edu/id/eprint/41145

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