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Volume 8
Issue 4
10.3390/axioms8040112
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Article

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

by
Irem Kucukoglu
1,
Burcin Simsek
2 and
Yilmaz Simsek
3,*
1
Department of Engineering Fundamental Sciences, Faculty of Engineering, Alanya Alaaddin Keykubat University, TR-07425 Antalya, Turkey
2
Department of Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA
3
Department of Mathematics, Faculty of Science University of Akdeniz, TR-07058 Antalya, Turkey
*
Author to whom correspondence should be addressed.
Axioms 2019, 8(4), 112; https://doi.org/10.3390/axioms8040112
Received: 14 September 2019 / Revised: 8 October 2019 / Accepted: 9 October 2019 / Published: 11 October 2019
(This article belongs to the Special Issue Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday)
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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–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-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. View Full-Text
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 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
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
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MDPI and ACS Style

Kucukoglu, I.; Simsek, B.; Simsek, Y. Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function. Axioms 2019, 8, 112. https://doi.org/10.3390/axioms8040112

AMA Style

Kucukoglu I, Simsek B, Simsek Y. Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function. Axioms. 2019; 8(4):112. https://doi.org/10.3390/axioms8040112

Chicago/Turabian Style

Kucukoglu, Irem, Burcin Simsek, and Yilmaz Simsek. 2019. "Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function" Axioms 8, no. 4: 112. https://doi.org/10.3390/axioms8040112

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