  We prove an equivalence between the existence of the first structure relation
satisfied 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-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 $\infty$.
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.
