Two-Variable Type 2 Poly-Fubini Polynomials

Ghulam Muhiuddin,Waseem Ahmad Khan,Ugur Duran

doi:10.3390/math9030281

Abstract

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

Highlights

IntroductionThe Bernoulli polynomials of the second kind are defined as follows (cf [5,11,12]):
Throughout the paper, we use N := {1, 2, 3, · · · } and N0 = N ∪ {0}
A relationship involving Stirling numbers of the first kind, the two-variable Fubini polynomials, and two-variable type 2 poly-Fubini polynomials is stated by the following theorem

Summary

Introduction

The Bernoulli polynomials of the second kind are defined as follows (cf [5,11,12]):. Using the polyexponential function Eik ( x ), Kim-Kim [12] considered type 2 poly-Bernoulli polynomials, given by. By utilizing the unipoly function uk ( x | p), Kim-Kim [12] defined unipoly-Bernoulli polynomials as follows:. We introduce a new extension of the two-variable Fubini polynomials by means of the polyexponential function, which we call two-variable type. We derive some useful relations including the Stirling numbers of the first and the second kinds, the usual Fubini polynomials, and the Bernoulli polynomials of higher-order. We introduce twovariable unipoly-Fubini polynomials via unipoly function and acquire diverse properties including derivative and integral properties. We provide some relationships covering the Stirling numbers of the first and the second kinds, the two-variable unipolyFubini polynomials, and the Daehee polynomials

Two-Variable Type 2 Poly-Fubini Polynomials and Numbers

Two-Variable Unipoly-Fubini Polynomials

Conclusions