A Novel Kind of the Type 2 Poly-Fubini Polynomials and Numbers

Abstract

Motivation by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim [9], in the present paper, we consider a new class of new generating function for the Fubini polynomials, called the type 2 poly-Fubini polynomials by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Fubini polynomials equal a linear combination of the classical of the Fubini polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Fubini polynomials and Bernoulli polynomials of order r. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim [9]. We introduce the type 2 unipoly-Fubini polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Fubini polynomials and the classical Fubini polynomials.

Unified degenerate Apostol-type Bernoulli, Euler, Genocchi, and Fubini polynomials

“…In recent years, many mathematicians ( [1,[4][5][6][7][8][9][10][11][12][13][14][15][16]) studied and gave some relations for the two-variable Fubini polynomials. Duran et al in [5] introduced q-Fubini polynomials and proved some theorems for the q-Fubini polynomials. )…”

“…As usual, throughout this paper, we always make use of the following notation; N denotes the set of natural numbers, N 0 denotes the set of nonnegative integers, R denotes the set of real numbers, and C denotes the set of complex numbers. We begin by introducing the following definitions and notations (see also [6,11,12,[16][17][18][19]).…”

“…where v ∈ N, a, b, β ∈ R and a = b. Two-variable Fubini polynomials of order α are defined in ( [1,[4][5][6][7][8][9][10][13][14][15]) as follows…”