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

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
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
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.

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