Investigating the Properties and Dynamic Applications of Δ_<i>h</i> Legendre–Appell Polynomials

Abstract

This research aims to introduce and examine a new type of polynomial called the Δh Legendre–Appell polynomials. We use the monomiality principle and operational rules to define the Δh Legendre–Appell polynomials and explore their properties. We derive the generating function and recurrence relations for these polynomials and their explicit formulas, recurrence relations, and summation formulas. We also verify the monomiality principle for these polynomials and express them in determinant form. Additionally, we establish similar results for the Δh Legendre–Bernoulli, Euler, and Genocchi polynomials.

Keywords

• Δ_<i>h</i> sequences
• monomiality principle
• Legendre–Appell polynomials
• explicit forms
• determinant form