Generating Functions for Special Polynomials and Numbers Including Apostol-Type and Humbert-Type Polynomials

Abstract

The aim of this paper is to give generating functions and to prove various properties for some new families of special polynomials and numbers. Several interesting properties of such families and their connections with other polynomials and numbers of the Bernoulli, Euler, Apostol–Bernoulli, Apostol–Euler, Genocchi and Fibonacci type are presented. Furthermore, the Fibonacci-type polynomials of higher order in two variables and a new family of special polynomials \((x,y)\mapsto \mathbb {G}_{d}(x,y;k,m,n)\), including several particular cases, are introduced and studied. Finally, a class of polynomials and corresponding numbers, obtained by a modification of the generating function of Humbert’s polynomials, is also considered.