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We prove an equivalence between the existence of the first structure relationsatisfied by a sequence of monic orthogonal polynomials$\{P_n\}_{n=0}^{\infty}$, the orthogonality of the second derivatives$\{\mathbb{D}_{x}^2P_n\}_{n= 2}^{\infty}$ and a generalized Sturm-Liouvilletype equation. Our treatment of the generalized Bochner theorem leads toexplicit solutions of the difference equation [Vinet L., Zhedanov A., J.Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monicorthogonal polynomials that satisfy the first structure relation are Wilsonpolynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials andtheir special or limiting cases as one or more parameters tend to $\infty$.This work extends our previous result [arXiv:1711.03349] concerning aconjecture due to Ismail. We also derive a second structure relation forpolynomials satisfying the first structure relation.

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