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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.

In this paper, we present linear differential equations for the generating functions of the Poisson-Charlier, actuarial, and Meixner polynomials. Also, we give an application for each case. [ABSTRACT FROM AUTHOR]

This paper deals with the hypergeometric orthogonal polynomials with special emphasis on the Hahn polynomials and the continuous dual Hahn polynomials. New representations, generating functions, and summation formulas are derived. The addition theorems for the Krawtchouk and the Meixner-Pollaczek polynomials are also included. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]