Nangho, Maurice KenfackJordaan, Kerstin Heidrun2019-08-122019-08-122018-11-27Kenfack, M. & Jordaan, K. 2018, 'Structure relations of classical orthogonal polynomials in the quadratic and q-quadratic variable', Symmetry, Integrability and Geometry: Methods and Applications, vol. 14, art. 126, pp. 1-26.1815-065910.3842/SIGMA.2018.126http://hdl.handle.net/2263/70950This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14).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.enThe authors retain the copyright for their papers published in SIGMA under the terms of the Creative Commons Attribution-ShareAlike License.Classical orthogonal polynomialsClassical q-orthogonal polynomialsAskey{ Wilson polynomialsWilson polynomialsStructure relationsCharacterization theoremsArticle