A structure relation for some specific orthogonal polynomials

TL;DR

This paper characterizes orthogonal polynomial sequences satisfying a specific difference relation on a q-quadratic lattice, deriving new structure relations for certain polynomial families.

Contribution

It provides a complete characterization of orthogonal polynomials satisfying a particular difference relation, leading to new structure relations for specific families.

Findings

Derived new structure relations for orthogonal polynomials.

Characterized all sequences satisfying the given difference relation.

Connected the relations to specific polynomial families.

Abstract

By characterizing all orthogonal polynomials sequences $(P_n)_{n\geq 0}$ for which $$ (ax+b)(\triangle +2\,\mathrm{I})P_n(x(s-1/2))=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\,\mathrm{I}$ is the identity operator, $x$ defines a $q$-quadratic lattice, $\triangle f(s)=f(s+1)-f(s)$, and $(a_n)_{n\geq0}$, $(b_n)_{n\geq0}$ and $(c_n)_{n\geq0}$ are sequences of complex numbers, we derive some new structure relations for some specific families of orthogonal polynomials.