Exploring Degenerate Peter–Genocchi Polynomials and Their Graphical Analysis

In this paper, we introduce and investigate a new class of special polynomials called degenerate Peter–Genocchi polynomials. We define these polynomials and their associated numbers via an explicit generating function and explore a variety of their fundamental algebraic and analytic properties. In particular, we derive summation formulas (expressing polynomials via their associated numbers), addition formulas (splitting the polynomial argument), an implicit summation formula (a bivariate convolution identity), and symmetric identities. We also establish connections with degenerate Stirling numbers of both kinds, higher-order degenerate Genocchi polynomials, and higher-order degenerate Daehee polynomials. Furthermore, we investigate derivative properties and present a differential operator formula. Finally, we provide tables of approximate zeros and graphical representations of the zeros in the complex plane.