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The Degenerate Three-Variable Hermite-Based Apostol-Frobenius-type Poly-Genocchi Polynomials with Parameters a and b

Roberto B. Corcino, Cristina B. Corcino

TL;DR

The paper develops a degenerate, three-variable Hermite-based family of Apostol-Frobenius-type poly-Genocchi polynomials with parameters $a$ and $b$, defined by a generating function that combines the degenerate polyexponential $Ei_{k,rho}$ with degenerate exponential factors in $x,y,z$. It provides formal definitions, specializations (alpha=1), and higher-order generalizations, along with addition formulas and polynomial representations. A key contribution is the connection to $r$-Whitney numbers of the first and second kinds, enabling combinatorial interpretations and reductions to known Genocchi-type polynomials under suitable parameter choices. These results extend the toolkit of Genocchi-type polynomials in degenerate and Apostol-Frobenius contexts with potential applications in number theory and combinatorics.

Abstract

In this paper, we introduce the degenerate three-variable Hermite-based Apostol{Frobenius-type poly-Genocchi polynomials by integrating the modified degenerate polyexponential function with three-variable Hermite polynomials and Frobenius polynomials. We investigate several fundamental properties of these polynomials and derive a variety of identities and formulas, including explicit formulas, addition formula and expression in polynomial form. Moreover, we establish meaningful connections between these polynomials and the r-Whitney numbers of both the first and second kinds.

The Degenerate Three-Variable Hermite-Based Apostol-Frobenius-type Poly-Genocchi Polynomials with Parameters a and b

TL;DR

The paper develops a degenerate, three-variable Hermite-based family of Apostol-Frobenius-type poly-Genocchi polynomials with parameters and , defined by a generating function that combines the degenerate polyexponential with degenerate exponential factors in . It provides formal definitions, specializations (alpha=1), and higher-order generalizations, along with addition formulas and polynomial representations. A key contribution is the connection to -Whitney numbers of the first and second kinds, enabling combinatorial interpretations and reductions to known Genocchi-type polynomials under suitable parameter choices. These results extend the toolkit of Genocchi-type polynomials in degenerate and Apostol-Frobenius contexts with potential applications in number theory and combinatorics.

Abstract

In this paper, we introduce the degenerate three-variable Hermite-based Apostol{Frobenius-type poly-Genocchi polynomials by integrating the modified degenerate polyexponential function with three-variable Hermite polynomials and Frobenius polynomials. We investigate several fundamental properties of these polynomials and derive a variety of identities and formulas, including explicit formulas, addition formula and expression in polynomial form. Moreover, we establish meaningful connections between these polynomials and the r-Whitney numbers of both the first and second kinds.
Paper Structure (4 sections, 4 theorems, 72 equations)

This paper contains 4 sections, 4 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Definition and Some Explicit Formulas
  3. Relation With Some Genocchi-type Polynomials
  4. Acknowledgement

Key Result

Theorem 2.2

The degenerate three-variable Hermite-based Apostol-Frobenius-type poly-Genocchi polynomials with parameters $a$ and $b$ are equal to where $\widehat{\mathcal{G}}_{0}^{(k,\alpha)}(x,y,z; \lambda, \rho, u, a, b)=0$, $A_n(u)$ is the Eulerian polynomial and $\mathcal{B}_{m_1,m_2,\ldots,m_{k-1}}(i,\rho-1)$ satisfies deg_bern-multi.

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof