Some Certain Classes of Combinatorial Numbers and Polynomials Attached to Dirichlet Characters: Their Construction by p-Adic Integration and Applications to Probability Distribution Functions

The aim of this chapter is to survey on old and new identities for some certain classes of combinatorial numbers and polynomials derived from the non-trivial Dirichlet characters and p-adic integrals. This chapter is especially motivated by the recent papers (Simsek, Turk J Math 42:557–577, 2018; Srivastava et al., J Number Theory 181:117–146, 2017; Kucukoglu et al. Turk J Math 43:2337–2353, 2019; Axioms 8(4):112, 2019) in which the aforementioned combinatorial numbers and polynomials were extensively investigated and studied in order to obtain new results. In this chapter, after recalling the origin of the aforementioned combinatorial numbers and polynomials, which goes back to the paper (Simsek, Turk J Math 42:557–577, 2018), a compilation has been made on what has been done from the paper (Simsek, Turk J Math 42:557–577, 2018) up to present days about the main properties and relations of these combinatorial numbers and polynomials. Moreover, with the aid of some known and new formulas, relations, and identities, which involve some kinds of special numbers and polynomials such as the Apostol-type, the Peters-type, the Boole-type numbers and polynomials the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Genocchi numbers and polynomials, the Stirling numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind), the binomial coefficients, the falling factorial, etc., we give further new formulas and identities regarding these combinatorial numbers and polynomials. Besides, some derivative and integral formulas, involving not only these combinatorial numbers and polynomials, but also their generating functions, are presented in addition to those given for their positive and negative higher-order extensions. By using Wolfram programming language in Mathematica, we present some plots for these combinatorial numbers and polynomials with their generating functions. Finally, in order to do mathematical analysis of the results in an interdisciplinary way, we present some observations on a few applications of the positive and negative higher-order extension of the generating functions for combinatorial numbers and polynomials to the probability theory for researchers to shed light on their future interdisciplinary studies.