On bivariate classical orthogonal polynomials

Marcellán, Francisco; Marriaga, Misael E.; Pérez, Teresa E.; Piñar, Miguel A.

doi:10.1016/j.amc.2017.12.040

Cited by 5 publications

(1 citation statement)

References 18 publications

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“…al. [14] proved that the moment functional is classical if and only if its moments satisfy two three-term relations. In this work using the vector notation of polynomials in two variables, we establish the following extension of (1) to classical orthogonal polynomials in two variables…”

Section: Introduction and Notations 1introductionmentioning

confidence: 99%

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

Nangho¹,

Jordaan²

2018

SIGMA

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We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials {P n } ∞ n=0 , the orthogonality of the second derivatives {D 2x P n } ∞ n=2 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 ∞. 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.

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Section: Introduction and Notations 1introductionmentioning

confidence: 99%