On Gould–Hopper-Based Fully Degenerate Poly-Bernoulli Polynomials with a <i>q</i>-Parameter

Abstract

We firstly consider the fully degenerate Gould⁻Hopper polynomials with a q parameter and investigate some of their properties including difference rule, inversion formula and addition formula. We then introduce the Gould⁻Hopper-based fully degenerate poly-Bernoulli polynomials with a q parameter and provide some of their diverse basic identities and properties including not only addition property, but also difference rule properties. By the same way of mentioned polynomials, we define the Gould⁻Hopper-based fully degenerate ( α , q ) -Stirling polynomials of the second kind, and then give many relations. Moreover, we derive multifarious correlations and identities for foregoing polynomials and numbers, including recurrence relations and implicit summation formulas.

Keywords

• Gould–Hopper polynomials
• Bernoulli polynomials
• Hermite polynomials
• poly Bernoulli polynomials
• Stirling numbers of second kind
• Polylogarithm functions
• Cauchy product