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.
