Construction of the type 2 poly-Frobenius–Genocchi polynomials with their certain applications

Abstract

Kim and Kim (Russ. J. Math. Phys. 26(1):40-49, 2019) have studied the type 2 poly-Bernoulli polynomials. Inspired by their work, we consider a new class of the Frobenius-Genocchi polynomials, which is called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. We also derive some new relations and properties including the Stirling numbers of the first and second kinds. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, motivated by the definition of the unipoly-Bernoulli polynomials given in (Kim and Kim in Russ. J. Math. Phys. 26(1):40-49, 2019), we introduce the unipoly-Frobenius-Genocchi polynomials via a unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

A Note on the Poly-Bernoulli Polynomials of the Second Kind

“…In addition, they defied the unipoly function, attached a arithmetic function p, and found some interesting identities related to Bernoulli numbers, poly-Bernoulli polynomials, and the Stirling numbers of the first kind and second kind. The polyexponential function have been used to define some special polynomials by some researcher and found many interesting identities of those polynomials (see [11,12,[20][21][22][23][24][25][26]).…”

“…By (7), we know that e 1 ðxÞ = e x . Recently, some authors applied the polyexponential functions and the polylogarithm functions to degenerate Bernoulli polynomials, type 2 poly-Apostol-Bernoulli polynomials, type 2 degenerate poly-Euler polynomials, and poly-Genocchi polynomials and found many interesting identities about those polynomials (see [11,12,[20][21][22][23][24][25][26]).…”

A New Family of Degenerate Poly-Genocchi Polynomials with Its Certain Properties

“…The degenerate Stirling numbers of the second kind [31] are given by (see [2,[13][14][15][16][17][18][19][20][21][22][25][26][27][28][29][30][31][32])…”

“…The classical Euler polynomials E n ðxÞ and the classical Genocchi polynomials G n ðxÞ are, respectively, defined by the following generating functions (see [12][13][14][15][16][17][18][19][20][21][22]):…”

Two-Variable Type 2 Poly-Fubini Polynomials

“…Some mathematicians have considered and examined several extensions of special polynomials via polyexponential function, cf. [5,11,13,16,17] and see also the references cited therein. For example, Duran et al [11] defined type 2 poly-Frobenius-Genocchi polynomials by the following Maclaurin series expansion (in a suitable neighborhood of z = 0):…”