On a moment generalization of some classical second-order differential equations generating classical orthogonal polynomials

TL;DR

This paper introduces new polynomial systems as solutions to generalized functional equations, extending classical orthogonal polynomials through fractional, q-difference, and other equations that converge to classical differential equations.

Contribution

It constructs novel polynomial families solving generalized functional equations that unify and extend classical orthogonal polynomials via fractional and q-difference equations.

Findings

New polynomial systems solving generalized functional equations

Convergence to classical orthogonal polynomials and differential equations

Unified framework for classical and generalized polynomial families

Abstract

The aim of the work is to construct new polynomial systems, which are solutions to certain functional equations which generalize the second-order differential equations satisfied by the so called classical orthogonal polynomial families of Jacobi, Laguerre, Hermite and Bessel. These functional equations can be chosen to be of different type: fractional differential equations, q-difference equations, etc, which converge to their respective differential equations of the aforesaid classical orthogonal polynomials. In addition to this, there exists a confluence of both the families of polynomials constructed and the functional equations who approach to the classical families of polynomials and second-order differential equations, respectively