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

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.