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Monic bivariate polynomials on quadratic and \(q\)-quadratic lattices. (English) Zbl 1510.33016

Summary: Monic families of bivariate Askey-Wilson polynomials and \(q\)-Racah polynomials are explicitly given. Monic families of bivariate orthogonal polynomials, both in quadratic and \(q\)-quadratic lattices, are also explicitly given using appropriate limit relations.

MSC:

33D50 Orthogonal polynomials and functions in several variables expressible in terms of basic hypergeometric functions in one variable
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33E30 Other functions coming from differential, difference and integral equations

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References:

[1] Area, I.; Atakishiyev, NM; Godoy, E.; Rodal, J., Linear partial \(q\)-difference equations on \(q\)-linear lattices and their bivariate \(q\)-orthogonal polynomial solutions, Appl. Math. Comput., 223, 520-536 (2013) · Zbl 1329.39011
[2] Area, I.; Godoy, E., On limit relations between some families of bivariate hypergeometric orthogonal polynomials, J. Phys. A Math. Theor., 46, 35202, 11 (2013) · Zbl 1272.33016
[3] Area, I.; Godoy, E.; Rodal, J., On a class of bivariate second-order linear partial difference equations and their monic orthogonal polynomial solutions, J. Math. Anal. Appl., 389, 165-178 (2012) · Zbl 1236.39015
[4] Area, I.; Godoy, E.; Ronveaux, A.; Zarzo, A., Corrigendum: minimal recurrence relations for connection coefficients between classical orthogonal polynomials: discrete case, J. Comput. Appl. Math., 89, 2, 309-325 (1998) · Zbl 0934.33014
[5] Area, I.; Godoy, E.; Ronveaux, A.; Zarzo, A., Bivariate second-order linear partial differential equations and orthogonal polynomial solutions, J. Math. Anal. Appl., 387, 2, 1188-1208 (2012) · Zbl 1248.35037
[6] Askey, R.; Wilson, J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Am. Math. Soc., 54, iv+55 (1985) · Zbl 0572.33012
[7] Atakishiyev, NM; Rahman, M.; Suslov, SK, On classical orthogonal polynomials, Constr. Approx., 11, 2, 181-226 (1995) · Zbl 0837.33010
[8] Dunkl, CF; Xu, Y., Orthogonal Polynomials of Several Variables, Volume 81 of Encyclopedia of Mathematics and its Applications (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0964.33001
[9] Foupouagnigni, M., On difference equations for orthogonal polynomials on nonuniform lattices, J. Differ. Equ. Appl., 14, 127-174 (2008) · Zbl 1220.33017
[10] Foupouagnigni, M.; Koepf, W.; Kenfack-Nangho, M.; Mboutngam, S., On solutions of holonomic divided-difference equations on nonuniform lattices, Axioms, 2, 404-434 (2013) · Zbl 1301.33024
[11] Geronimo, JS; Iliev, P., Bispectrality of multivariable Racah-Wilson polynomials, Constr. Approx., 31, 417-457 (2010) · Zbl 1208.47034
[12] Ismail, MEH; Stanton, D., Some combinatorial and analytical identities, Ann. Comb., 16, 755-771 (2012) · Zbl 1256.05021
[13] Koekoek, R.; Lesky, PA; Swarttouw, RF, Hypergeometric Orthogonal Polynomials and Their \(q\)-Analogues, Springer Monographs in Mathematics (2010), Berlin: Springer, Berlin · Zbl 1200.33012
[14] Gasper, G.; Rahman, M., Some systems of multivariable orthogonal Askey-Wilson polynomials, in theory and applications of special functions, Dev. Math., 13, 209-219 (2005) · Zbl 1219.33027
[15] Gasper, G.; Rahman, M., Some systems of multivariable orthogonal \(q\)-Racah polynomials, Ramanujan J., 13, 389-405 (2007) · Zbl 1121.33019
[16] Godoy, E.; Ronveaux, A.; Zarzo, A.; Area, I., Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case, J. Comput. Appl. Math., 84, 2, 257-275 (1997) · Zbl 0909.65008
[17] Iliev, P., Bispectral commuting difference operators for multivariable Askey-Wilson polynomials, Trans. Am. Math. Soc., 363, 3, 1577-1598 (2011) · Zbl 1215.39009
[18] Lewanowicz, S.; Woźny, P., Two-variable orthogonal polynomials of big \(q\)-Jacobi type, J. Comput. Appl. Math., 233, 1554-1561 (2010) · Zbl 1190.33018
[19] Lewanowicz, S.; Woźny, P.; Nowak, R., Structure relations for the bivariate big \(q\)-Jacobi polynomials, App. Math. Comput., 219, 8790-8802 (2013) · Zbl 1306.33022
[20] Magnus, A.P.: Associated Askey-Wilson polynomials as Laguerre-Hahn orthogonal polynomials, In: Orthogonal polynomials and their applications (Segovia, 1986), Lecture Notes in Mathematics, vol. 1329, pp. 261-278. Springer, Berlin (1988) · Zbl 0645.33015
[21] Magnus, A.P.: Special nonuniform lattice (SNUL) orthogonal polynomials on discrete dense sets of points, In: Proceedings of the International Conference on Orthogonality, Moment Problems and Continued Fractions (Delft, 1994), vol. 65, pp. 253-265 (1995) · Zbl 0847.33008
[22] Nikiforov, AF; Suslov, SK; Uvarov, VB, Classical Orthogonal Polynomials of a Discrete Variable. Springer Series in Computational Physics (1991), Berlin: Springer, Berlin · Zbl 0743.33001
[23] Njionou Sadjang, P.; Koepf, W.; Foupouagnigni, M., On structure formulas for Wilson polynomials, Int. Transf. Spec. Funct., 26, 12, 1000-1014 (2015) · Zbl 1331.33011
[24] Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Department of Commerce and Cambridge University Press, National Institute of Standards and Technology U.S (2010) · Zbl 1198.00002
[25] Post, S., Racah polynomials and recoupling schemes of \(\varvec{\mathfrak{su}(1,1)} \), SIGMA, 11, 057 (2015) · Zbl 1325.33009
[26] Rodal, J.; Area, I.; Godoy, E., Orthogonal polynomials of two discrete variables on the simplex, Integral Transforms Spec Funct, 16, 263-280 (2005) · Zbl 1065.33013
[27] Tcheutia, DD; Guemo Tefo, Y.; Foupouagnigni, M.; Godoy, E.; Area, I., Linear partial divided-difference equation satisfied by multivariate orthogonal polynomials on quadratic lattices, Math. Model. Nat. Phenom., 12, 3, 14-43 (2017) · Zbl 1384.33023
[28] Tcheutia, DD; Foupouagnigni, M.; Guemo Tefo, Y.; Area, I., Divided-difference equation and three-term recurrence relations of some systems of bivariate \(q\)-orthogonal polynomials, J. Differ. Equ. Appl., 23, 12, 2004-2036 (2017) · Zbl 1388.39004
[29] Tratnik, MV, Some multivariable orthogonal polynomials of the Askey tableau-continuous families, J. Math. Phys., 32, 2065-2073 (1991) · Zbl 0746.33007
[30] Tratnik, MV, Some multivariable orthogonal polynomials of the Askey tableau-discrete families, J. Math. Phys., 32, 2337-2342 (1991) · Zbl 0742.33007
[31] Witte, N.S.: Semi-classical orthogonal polynomial systems on non-uniform lattices, deformations of the Askey table and analogs of isomonodromy. http://arxiv.org/abs/1204.2328 (2011)
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