Structure relations of classical orthogonal polynomials in the quadratic and q-quadratic variable

Abstract

We prove an equivalence between the existence of the rst structure relation satis ed by a sequence of monic orthogonal polynomials fPng1n =0, the orthogonality of the second derivatives fD2 xPng1n =2 and a generalized Sturm{Liouville type equation. Our treat- ment of the generalized Bochner theorem leads to explicit solutions of the di erence equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45{56], which proves that the only monic orthogonal polynomials that satisfy the rst structure relation are Wilson poly- nomials, continuous dual Hahn polynomials, Askey{Wilson polynomials and their special or limiting cases as one or more parameters tend to 1. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the rst structure relation.