Taiwanese Journal of Mathematics

Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

Samuel G. Moreno and Esther M. García-Caballero

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We consider the continuous $q$-Jacobi polynomials $\{P_n^{(\alpha,\beta)}(\cdot|q)\}_{n=0}^{\infty}$, extending the variable and the parameters beyond classical considerations. For those new allowed values of the parameters for which Favard's theorem fails to work, we construct inner products in which the presence of the Askey-Wilson divided difference operator provides the $q$-Sobolev character of the non-standard orthogonality for the corresponding family.

Article information

Taiwanese J. Math., Volume 15, Number 4 (2011), 1677-1690.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 33D45: Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]

continuous $q$-Jacobi polynomials non-standard orthogonality Favard's theorem


Moreno, Samuel G.; García-Caballero, Esther M. Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials. Taiwanese J. Math. 15 (2011), no. 4, 1677--1690. doi:10.11650/twjm/1500406372. https://projecteuclid.org/euclid.twjm/1500406372

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  • M. Alfaro, M. Álvarez de Morales and M. L. Rezola, Orthogonality of the Jacobi polynomials with negative integer parameters, J. Comput. Appl. Math., 145 (2002), 379-386.
  • M. Alfaro, T. E. Pérez, M. A. Piñar and M. L. Rezola, Sobolev orthogonal polynomials: the discrete-continuous case, Meth. Appl. Anal., 6 (1999), 593-616.
  • M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux, Non-standard orthogonality for Meixner polynomials, ETNA, Electron. Trans. Numer. Anal., 9 (1999), 1-25.
  • M. Álvarez de Morales, T. E. Pérez and M. A. Piñar, Sobolev orthogonality for the Gegenbauer polynomials $\{C_n^{(-N+1/2)}\}_{n\geq 0}$, J. Comput. Appl. Math., 100 (1998), 111-120.
  • T. K. Araaya, The Meixner-Pollaczek polynomials and a system of orthogonal polynomials in a strip, J. Comput. Appl. Math., 170 (2004), 241-254.
  • T. K. Araaya, The symmetric Meixner-Pollaczek polynomials with real parameter, J. Math. Anal. Appl., 305 (2005), 411-423.
  • R. Askey and J. Wilson, Some basic hypergeometric polynomials that generalize Jacobi polynomials, Memoirs Amer. Math. Soc., 319, Providence, Rhode Island, 1985.
  • R. S. Costas-Santos and J. F. Sánchez-Lara, Extensions of discrete classical orthogonal polynomials beyond the orthogonality, J. Comput. Appl. Math., 225 (2009), 440-451.
  • D. Dominici, Some remarks on a paper by L. Carlitz, J. Comput. Appl. Math., 198 (2007), 129-142.
  • G. Gasper and M. Rahman, Basic hypergeometric series, second ed., Cambridge Univ. Press, 2004.
  • Samuel G. Moreno and E. M. Garc\ipz a-Caballero, Linear interpolation and Sobolev orthogonality, J. Approx. Theory, 161 (2008), 35-48.
  • Samuel G. Moreno and E. M. Garc\ipz a-Caballero, The orthogonality of the Meixner-Pollaczek polynomials $\{P_n^{({\scriptscriptstyle \frac{1-N}{2}})}(\cdot;\phi)\}_{n=0}^{\infty}$ for positive integers $N$, submitted for publication.
  • Samuel G. Moreno and E. M. Garc\ipz a-Caballero, Non-classical orthogonality relations for big and little $q$-Jacobi polynomials, J. Approx. Theory, 162 (2010), 303-322.
  • Samuel G. Moreno and E. M. Garc\ipz a-Caballero, New orthogonality relations for the continuous and the discrete $q$-ultraspherical polynomials, J. Math. Anal. Appl., 369 (2010), 386-399.
  • K. H. Kwon and L. L. Littlejohn, The orthogonality of the Laguerre polynomials $\{L^{-k}_n(x)\}$ for positive integers $k$, Ann. Numer. Math., 2 (1995), 289-303.
  • K. H. Kwon and L. L. Littlejohn, Sobolev orthogonal polynomials and second-order differential equations, Rocky Mountain J. Math., 28 (1998), 547-594.
  • R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, Technical Report 98-17, Delft University of Technology, 1998.
  • T. E. Pérez and M. A. Piñar, On Sobolev orthogonality for the generalized Laguerre polynomials, J. Approx. Theory, 86 (1996), 278-285.
  • M. Rahman, The linearization of the product of continuous $q$-Jacobi polynomials, Canad. J. Math., 33 (1981), 961-987.
  • V. N. Singh, The basic analogues of identities of the Cayley-Orr type, J. London Math. Soc., 34 (1959), 15-22.