Peters type polynomials and numbers and their generating functions: Approach with p-adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p-adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well-known special numbers and polynomials are presented. By using p-adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol-type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well-known formulas. Finally, two open problems for interpolation functions for Apostol-type Peters numbers and polynomials are revealed.