### 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.

pdf (508 kb)   tex (26 kb)       [previous version:  pdf (485 kb)   tex (26 kb)]

References

1. Al-Salam W.A., Characterization theorems for orthogonal polynomials, in Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, 1-24.
2. Al-Salam W.A., Chihara T.S., Another characterization of the classical orthogonal polynomials, SIAM J. Math. Anal. 3 (1972), 65-70.
3. Álvarez Nodarse R., On characterizations of classical polynomials, J. Comput. Appl. Math. 196 (2006), 320-337.
4. Andrews G.E., Askey R., Classical orthogonal polynomials, in Orthogonal Polynomials and Applications (Bar-le-Duc, 1984), Lecture Notes in Math., Vol. 1171, Springer, Berlin, 1985, 36-62.
5. Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), iv+55 pages.
6. Atakishiyev N.M., Rahman M., Suslov S.K., On classical orthogonal polynomials, Constr. Approx. 11 (1995), 181-226.
7. Bochner S., Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730-736.
8. Costas-Santos R.S., Marcellán F., $q$-classical orthogonal polynomials: a general difference calculus approach, Acta Appl. Math. 111 (2010), 107-128, math.CA/0612097.
9. Datta S., Griffin J., A characterization of some $q$-orthogonal polynomials, Ramanujan J. 12 (2006), 425-437.
10. Foupouagnigni M., On difference equations for orthogonal polynomials on nonuniform lattices, J. Difference Equ. Appl. 14 (2008), 127-174.
11. García A.G., Marcellán F., Salto L., A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math. 57 (1995), 147-162.
12. Grünbaum F.A., Haine L., The $q$-version of a theorem of Bochner, J. Comput. Appl. Math. 68 (1996), 103-114.
13. Ismail M.E.H., A generalization of a theorem of Bochner, J. Comput. Appl. Math. 159 (2003), 319-324.
14. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
15. Jooste A., Zeros of Jacobi, Meixner and Krawtchouk polynomials, Ph.D. Thesis, University of Pretoria, 2013, available at https://repository.up.ac.za/handle/2263/30787.
16. Kenfack Nangho M., Foupouagnigni M., Koepf W., On exponential and trigonometric functions on nonuniform lattices, Ramanujan J., to appear.
17. Kenfack Nangho M., Jordaan K., A characterization of Askey-Wilson polynomials, Proc. Amer. Math. Soc., to appear, arXiv:1711.03349.
18. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
19. Koepf W., Schmersau D., Representations of orthogonal polynomials, J. Comput. Appl. Math. 90 (1998), 57-94, math.CA/9703217.
20. Koepf W., Schmersau D., On a structure formula for classical $q$-orthogonal polynomials, J. Comput. Appl. Math. 136 (2001), 99-107.
21. Koornwinder T.H., The structure relation for Askey-Wilson polynomials, J. Comput. Appl. Math. 207 (2007), 214-226, math.CA/0601303.
22. Magnus A.P., Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, in Orthogonal Polynomials and their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, Springer, Berlin, 1988, 261-278.
23. Marcellán F., Branquinho A., Petronilho J., Classical orthogonal polynomials: a functional approach, Acta Appl. Math. 34 (1994), 283-303.
24. Maroni P., Variations around classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math. 48 (1993), 133-155.
25. Maroni P., Semi-classical character and finite-type relations between polynomial sequences, Appl. Numer. Math. 31 (1999), 295-330.
26. Nikiforov A.F., Suslov S.K., Uvarov V.B., Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991.
27. Tcheutia D.D., Jooste A.S., Koepf W., Mixed recurrence equations and interlacing properties for zeros of sequences of classical $q$-orthogonal polynomials, Appl. Numer. Math. 125 (2018), 86-102.
28. Vinet L., Zhedanov A., Generalized Bochner theorem: characterization of the Askey-Wilson polynomials, J. Comput. Appl. Math. 211 (2008), 45-56, arXiv:0712.0069.