Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 126, 26 pages      arXiv:1801.10554      https://doi.org/10.3842/SIGMA.2018.126
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

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

Maurice Kenfack Nangho ab and Kerstin Jordaan c
a) Department of Mathematics and Applied Mathematics, University of Pretoria, Private bag X20 Hatfield, 0028 Pretoria, South Africa
b) Department of Mathematics and Computer Science, University of Dschang, Cameroon
c) Department of Decision Sciences, University of South Africa, PO Box 392, Pretoria, 0003, South Africa

Received January 31, 2018, in final form November 13, 2018; Published online November 27, 2018; Errors and misprints corrected April 01, 2022

Abstract
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 $\big\{\mathbb{D}_{x}^2P_n\big\}_{n= 2}^{\infty}$ 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.

Key words: classical orthogonal polynomials; classical $q$-orthogonal polynomials; Askey-Wilson polynomials; Wilson polynomials; structure relations; characterization theorems.

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