On a certain type of deformation of umbral orthogonal polynomials

TL;DR

This paper explores a novel deformation of umbral orthogonal polynomials by deriving complex polynomial families from foundational ones, revealing the underlying mechanisms of classical and hypergeometric orthogonal polynomials.

Contribution

It introduces a new deformation approach for umbral orthogonal polynomials and provides a formal framework to understand their classical and hypergeometric counterparts.

Findings

Derived complex families of orthogonal polynomials from basic ones.

Unveiled the mechanisms behind classical and hypergeometric orthogonal polynomials.

Established a formalism connecting known and new polynomial families.

Abstract

In the present paper we derive complicated families of orthogonal polynomials in one variable from scratch using the known ones as building blocks. We recall the basics of operational formalism and introduce the notations we use throughout this paper. Then we establish through this formalism the known families of orthogonal polynomials in order to see the mechanism that lies at the heart of classical and hypergeometric $q=1$ orthogonal polynomials and their associated families, which arise form this viewpoint in a very natural manner.