\(\Delta _h\)-Appell versions of U-Bernoulli and U-Euler polynomials: properties, zero distribution patterns, and the monomiality principle

Abstract

This paper explores the properties, generating functions, recurrence relations, and summation formulas of a novel class of polynomials, referred to as \(\Delta _h\)-type U-Bernoulli and \(\Delta _h\)-type U-Euler polynomials. We delve into the characterization of these polynomials, including the monomiality principle, and derive the corresponding derivative and multiplicative operators. Additionally, we provide computational values in tables and visually appealing representations of the zeros of these polynomials in figures, offering a comprehensive understanding of their behavior.