×

zbMATH — the first resource for mathematics

Orthogonality of \(q\)-polynomials for non-standard parameters. (English) Zbl 1229.33016
The paper presents a general tool, which provides a non-standard orthogonality property for any sequence of orthogonal polynomials \(\{p_n\}\) with some vanishing coefficient \(\gamma_n\) in the three-term recurrence relation:
\[ xp_n= p_{n+1} + \beta_n p_n + \gamma_n p_{n-1},\quad n \geq 0. \] The tool is a degenerate version of the Favard theorem, and it is applied to obtain the orthogonality of \(q\)-polynomials for non-standard parameters. Indeed, orthogonality for the Askey-Wilson polynomials \(p_n(x;a,b,c,d;q)\) is known only when the product of any two parameters \(a,b,c,d\) is not a negative integer power of \(q\), and the orthogonality of the big \(q\)-Jacobi polynomials \(p_n(x;a,b,c;q)\) is known when not any of the numbers \(a,b,c,abc^{-1}\) is a negative integer power of \(q\). With the help of the technique mentioned before, the orthogonality properties of the Askey-Wilson and big \(q\)-Jacobi polynomials for almost all values of the parameters are obtained.
Finally, the results are also applied to derive non-standard orthogonality properties for continuous dual \(q\)-Hahn, big \(q\)-Laguerre, \(q\)-Meixner and little \(q\)-Jacobi polynomials for non-classical parameters.

MSC:
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alfaro, M.; Álvarez de Morales, M.; Rezola, M.L., Orthogonality of the Jacobi polynomials with negative integer parameters, J. comput. appl. math., 145, 2, 379-386, (2002) · Zbl 1002.42016
[2] Alfaro, M.; Pérez, T.E.; Piñar, M.A.; Rezola, M.L., Sobolev orthogonal polynomials: the discrete-continuous case, Methods appl. anal., 6, 593-616, (1999) · Zbl 0980.42017
[3] Álvarez de Morales, M.; Pérez, T.E.; Piñar, M.A., Sobolev orthogonality for the Gegenbauer polynomials \(\{C_n^{(- N + 1 / 2)} \}_{n \geq 0}\), J. comput. appl. math., 100, 111-120, (1998) · Zbl 0931.33008
[4] Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, () · Zbl 0572.33012
[5] Atakishiyev, N.M.; Suslov, S.K., On the askey – wilson polynomials, Constr. approx., 8, 363-369, (1992) · Zbl 0762.33004
[6] Atkinson, F.V., Discrete and continuous boundary problems, () · Zbl 0117.05806
[7] Boas, R.P.; Buck, R.C., Polynomial expansions of analytic functions, (1964), Springer-Verlag Berlin · Zbl 0116.28105
[8] Chihara, T.S., An introduction to orthogonal polynomials, (1978), Gordon and Breach Science Publishers New York · Zbl 0389.33008
[9] Costas-Santos, R.S.; Sánchez-Lara, J.F., Extensions of discrete classical orthogonal polynomials beyond the orthogonality, J. comput. appl. math., 225, 2, 440-451, (2009) · Zbl 1167.42008
[10] Costas-Santos, R.S.; Marcellán, F., \(q\)-classical orthogonal polynomial: A general difference calculus approach, Acta appl. math., 111, 107-128, (2010) · Zbl 1204.33011
[11] Gasper, G.; Rahman, M., Basic hypergeometric series. encyclopedia of mathematics and its applications, vol. 35, (1990), Cambridge University Press Cambridge
[12] Koekoek, R.; Lesky, P.A.; Swarttouw, R.F., ()
[13] Kuijlaars, A.B.J.; Martinez-Finkelshtein, A.; Orive, R., Orthogonality of Jacobi polynomials with general parameters, Electron. trans. numer. anal., 19, 1-17, (2005) · Zbl 1075.33005
[14] Kwon, K.H.; Littlejohn, L.L., The orthogonality of the Laguerre polynomials \(\{L_n^{(- k)}(x) \}\) for a positive integer \(k\), Ann. numer. math., 2, 289-304, (1995) · Zbl 0831.33003
[15] Marcellán, F.; Álvarez-Nodarse, R., On the “favard theorem” and its extensions, J. comput. appl. math., 127, 1-2, 231-254, (2001) · Zbl 0970.33008
[16] Moreno, S.G.; García-Caballero, E.M., Non-standard orthogonality for the little \(q\)-Laguerre polynomials, Appl. math. lett., 22, 1745-1749, (2009) · Zbl 1177.33022
[17] Moreno, S.G.; García-Caballero, E.M., Non-classical orthogonality relations for big and little \(q\)-Jacobi polynomials, J. approx. theory, 162, 2, 303-322, (2010) · Zbl 1189.33031
[18] Moreno, S.G.; García-Caballero, E.M., New orthogonality relations for the continuous and the discrete \(q\)-ultraspherical polynomials, J. math. anal. appl., 369, 386-399, (2010) · Zbl 1247.33032
[19] S.G. Moreno, E.M. García-Caballero, Non-classical orthogonality relations for continuous \(q\)-Jacobi polynomials, Taiwan. J. Math. (in press). · Zbl 1237.33010
[20] Pérez, T.E.; Piñar, M.A., On Sobolev orthogonality for the generalized Laguerre polynomials, J. approx. theory, 86, 3, 278-285, (1996) · Zbl 0864.33009
[21] Spiridonov, V.; Zhedanov, A., Zeros and orthogonality of the askey – wilson polynomials for \(q\) a root of unity, Duke math. J., 89, 2, 283-305, (1997) · Zbl 0882.33007
[22] Vilenkin, N.Ja.; Klimyk, A.U., Representations of Lie groups and special functions. volume I, II, III, (1992), Kluwer Academic Publishers Dordrecht, The Netherlands · Zbl 0826.22001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.