## A characterization of Askey-Wilson polynomials

HTML articles powered by AMS MathViewer

- by Maurice Kenfack Nangho and Kerstin Jordaan PDF
- Proc. Amer. Math. Soc.
**147**(2019), 2465-2480 Request permission

## Abstract:

We show that the only monic orthogonal polynomials $\{P_n\}_{n=0}^{\infty }$ that satisfy \begin{equation*} \pi (x)\mathcal {D}_{q}^2P_{n}(x)=\sum _{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\; x=\cos \theta ,\;~ a_{n,n-2}\neq 0,~ n=2,3,\dots , \end{equation*} where $\pi (x)$ is a polynomial of degree at most $4$ and $\mathcal {D}_{q}$ is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases as one or more parameters tends to $\infty$. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.## References

- W. A. Al-Salam and T. S. Chihara,
*Another characterization of the classical orthogonal polynomials*, SIAM J. Math. Anal.**3**(1972), 65–70. MR**316772**, DOI 10.1137/0503007 - Richard Askey and James Wilson,
*Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials*, Mem. Amer. Math. Soc.**54**(1985), no. 319, iv+55. MR**783216**, DOI 10.1090/memo/0319 - N. M. Atakishiyev, M. Rahman, and S. K. Suslov,
*On classical orthogonal polynomials*, Constr. Approx.**11**(1995), no. 2, 181–226. MR**1342384**, DOI 10.1007/BF01203415 - S. Bochner,
*Über Sturm-Liouvillesche Polynomsysteme*, Math. Z.**29**(1929), no. 1, 730–736 (German). MR**1545034**, DOI 10.1007/BF01180560 - R. S. Costas-Santos and F. Marcellán,
*$q$-classical orthogonal polynomials: a general difference calculus approach*, Acta Appl. Math.**111**(2010), no. 1, 107–128. MR**2653053**, DOI 10.1007/s10440-009-9536-z - K. Driver and K. Jordaan,
*Bounds for extreme zeros of some classical orthogonal polynomials*, J. Approx. Theory**164**(2012), no. 9, 1200–1204. MR**2948561**, DOI 10.1016/j.jat.2012.05.014 - Somjit Datta and James Griffin,
*A characterization of some $q$-orthogonal polynomials*, Ramanujan J.**12**(2006), no. 3, 425–437. MR**2293799**, DOI 10.1007/s11139-006-0152-5 - J. Favard,
*Sur les polynomes de Tchebicheff*, CR Acad. Sci. Paris,**200**(1935), 2052–2055. - Mama Foupouagnigni,
*On difference equations for orthogonal polynomials on nonuniform lattices*, J. Difference Equ. Appl.**14**(2008), no. 2, 127–174. MR**2383000**, DOI 10.1080/10236190701536199 - M. Foupouagnigni, M. Kenfack Nangho, and S. Mboutngam,
*Characterization theorem for classical orthogonal polynomials on non-uniform lattices: the functional approach*, Integral Transforms Spec. Funct.**22**(2011), no. 10, 739–758. MR**2836867**, DOI 10.1080/10652469.2010.546996 - A. G. García, F. Marcellán, and L. Salto,
*A distributional study of discrete classical orthogonal polynomials*, Proceedings of the Fourth International Symposium on Orthogonal Polynomials and their Applications (Evian-Les-Bains, 1992), 1995, pp. 147–162. MR**1340932**, DOI 10.1016/0377-0427(93)E0241-D - F. Alberto Grünbaum and Luc Haine,
*The $q$-version of a theorem of Bochner*, J. Comput. Appl. Math.**68**(1996), no. 1-2, 103–114. MR**1418753**, DOI 10.1016/0377-0427(95)00262-6 - Wolfgang Hahn,
*Über die Jacobischen Polynome und zwei verwandte Polynomklassen*, Math. Z.**39**(1935), no. 1, 634–638 (German). MR**1545524**, DOI 10.1007/BF01201380 - Wolfgang Hahn,
*Über Polynome, die gleichzeitig zwei verschiedenen Orthogonalsystemen angehören*, Math. Nachr.**2**(1949), 263–278 (German). MR**32858**, DOI 10.1002/mana.19490020504 - Mourad E. H. Ismail,
*A generalization of a theorem of Bochner*, J. Comput. Appl. Math.**159**(2003), no. 2, 319–324. MR**2005962**, DOI 10.1016/S0377-0427(03)00536-3 - Mourad E. H. Ismail and Dennis Stanton,
*Applications of $q$-Taylor theorems*, Proceedings of the Sixth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Rome, 2001), 2003, pp. 259–272. MR**1985698**, DOI 10.1016/S0377-0427(02)00644-1 - Mourad E. H. Ismail,
*Classical and quantum orthogonal polynomials in one variable*, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR**2191786**, DOI 10.1017/CBO9781107325982 - F. H. Jackson,
*$q$-Difference Equations*, Amer. J. Math.**32**(1910), no. 4, 305–314. MR**1506108**, DOI 10.2307/2370183 - Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw,
*Hypergeometric orthogonal polynomials and their $q$-analogues*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR**2656096**, DOI 10.1007/978-3-642-05014-5 - Tom H. Koornwinder,
*The structure relation for Askey-Wilson polynomials*, J. Comput. Appl. Math.**207**(2007), no. 2, 214–226. MR**2345243**, DOI 10.1016/j.cam.2006.10.015 - Pascal Maroni,
*Une caractérisation des polynômes orthogonaux semi-classiques*, C. R. Acad. Sci. Paris Sér. I Math.**301**(1985), no. 6, 269–272 (French, with English summary). MR**803215** - P. Maroni,
*Prolégomènes à l’étude des polynômes orthogonaux semi-classiques*, Ann. Mat. Pura Appl. (4)**149**(1987), 165–184 (French, with English summary). MR**932783**, DOI 10.1007/BF01773932 - Luc Vinet and Alexei Zhedanov,
*Generalized Bochner theorem: characterization of the Askey-Wilson polynomials*, J. Comput. Appl. Math.**211**(2008), no. 1, 45–56. MR**2386827**, DOI 10.1016/j.cam.2006.11.004

## Additional Information

**Maurice Kenfack Nangho**- Affiliation: Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0003, South Africa
- Address at time of publication: Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon
- Email: Maurice.KenfackNangho@up.ac.za, maurice.kenfack@univ-dschang.org
**Kerstin Jordaan**- Affiliation: Department of Decision Sciences, University of South Africa, Pretoria, 0003, South Africa
- MR Author ID: 701645
- Email: jordakh@unisa.ac.za
- Received by editor(s): December 4, 2017
- Received by editor(s) in revised form: June 28, 2018
- Published electronically: March 1, 2019
- Additional Notes: The research of the first author was supported by a Vice-Chancellor’s Postdoctoral Fellowship from the University of Pretoria.

The second author served as corresponding author for this paper. The research by the second author was partially supported by the National Research Foundation of South Africa under grant number 108763. - Communicated by: Mourad Ismail
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**147**(2019), 2465-2480 - MSC (2010): Primary 33D45; Secondary 33C45
- DOI: https://doi.org/10.1090/proc/14317
- MathSciNet review: 3951425