Degenerate Sheffer-type polynomials and degenerate Sheffer polynomials associated with a random variable

TL;DR

This paper introduces and explores properties of degenerate Sheffer-type polynomials, including those associated with random variables, extending classical polynomials and analyzing their behavior for specific distributions.

Contribution

It develops the theory of degenerate Sheffer polynomials and their association with random variables, providing new results for higher-order degenerate Bernoulli and Euler polynomials.

Findings

Derived properties of degenerate Sheffer-type polynomials.

Introduced degenerate Sheffer polynomials linked to random variables.

Presented new results for higher-order degenerate Bernoulli and Euler polynomials.

Abstract

This paper has two primary contributions. First, we explore degenerate Sheffer-type polynomials, a hybrid of higher-order degenerate Bernoulli and Euler polynomials, and derive their properties. Second, assuming that the moment generating function of Y exists in a neighborhood of the origin, we introduce the degenerate Sheffer polynomials associated with Y. We then investigate their properties in general and for the specific cases of uniform and Bernoulli random variables. We also present new results for the higher-order degenerate Bernoulli and Euler polynomials.