A structure relation for some specific orthogonal polynomials

Abstract

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