Structure Relations of Classical Orthogonal Polynomials in the Quadratic and $q$-Quadratic Variable

We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials $\{P_n{\}}_{n=0}^{\mathrm{\infty }}$, the orthogonality of the second derivatives $\{{\mathbb{D}}_x^2{P}_n{\}}_{n=2}^{\mathrm{\infty }}$ and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to $\mathrm{\infty }$. 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 first structure relation.