Rocky Mountain Journal of Mathematics

On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

J.F. Sánchez-Lara

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Abstract

The discrete part of the discrete-continuous orthogonality \[ \mathscr {B}(f,g)=\mathscr {B}_d( f,g)+\mathscr {B}_c(f^{(N)},g^{(N)}), \] is studied for families of classical orthogonal polynomials such that the associated three-term recurrence relation \[ xp_n=p_{n+1}+\beta _np_n+ \gamma _n p_{n-1}, \] presents one vanishing coefficient $\gamma _n$, as in the case of Laguerre polynomials $L_n^{(-N)}$, Jacobi polynomials $P_n^{(-N,\beta )}$ and Gegenbauer polynomials $C_n^{(-N+1/2)}$ with $N\in \mathbb {N}$. It is shown that the discrete bilinear functional $\mathscr {B}_d$ can be replaced by a linear functional, $\mathscr {L}$, or by another bilinear functional related with $\mathscr {L}$, which allows us to reformulate the orthogonality in a much simpler way in the case of Laguerre polynomials and in a totally explicit form in the case of Jacobi and Gegenbauer polynomials.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 1 (2017), 267-288.

Dates
First available in Project Euclid: 3 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1488531900

Digital Object Identifier
doi:10.1216/RMJ-2017-47-1-267

Mathematical Reviews number (MathSciNet)
MR3619764

Zentralblatt MATH identifier
1360.42016

Subjects
Primary: 42C05: Orthogonal functions and polynomials, general theory [See also 33C45, 33C50, 33D45]
Secondary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

Keywords
Classical orthogonal polynomials Sobolev orthogonality

Citation

Sánchez-Lara, J.F. On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters. Rocky Mountain J. Math. 47 (2017), no. 1, 267--288. doi:10.1216/RMJ-2017-47-1-267. https://projecteuclid.org/euclid.rmjm/1488531900


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References

  • M. Alfaro, M. Álvarez de Morales and M.L. Rezola, Orthogonality of the Jacobi polynomials with negative integer parameters, J. Comp. 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 and M.A. Piñar, Sobolev orthogonality for the Gegenbauer polynomials $\{C_n^{(-N+1/2)}\}_{n\ge 0}$, J. Comp. Appl. Math. 100 (1998), 111–120.
  • G.E. Andrews, R. Askey and R. Roy, Special functions, Encycl. Math. Appl. 71, Cambridge Univ. Press, 1999.
  • A. Bruder and L.L. Littlejohn, Nonclassical Jacobi polynomials and Sobolev orthogonality, Res. Math. 61 (2012), 283–313.
  • T.S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978.
  • R.S. Costas-Santos and J.F. Sanchez-Lara, Extensions of discrete classical orthogonal polynomials beyond the orthogonality, J. Comp. Appl. Math. 225 (2009), 440–451.
  • ––––, Orthogonality of $q$-polynomials for nonstandard parameters, J. Appr. Th. 163 (2011), 1246–1268.
  • R.D. Costin, Orthogonality of Jacobi and Laguerre polynomials for general parameters via the Hadamard finite part, J. Appr. Th. 162 (2010), 141–152.
  • A. Draux, Polynômes orthogonaux formels. Applications. Lect. Notes Math. 974, Springer-Verlag, Berlin, 1983.
  • A. Erdélyi, et al., Higher transcendental functions, Bateman Manuscript Project 1, McGraw-Hill Book Company, New York, 1953.
  • W.N. Everitt, L.L. Littlejohn and R. Wellman, The Sobolev orthogonality and spectral analysis of the Laguerre polynomials $L_n^{-k}$ for positive integers $k$, J. Comp. Appl. Math. 171 (2004), 199–234.
  • I.H. Jung, K.H. Kwon and J.K. Lee, Sobolev orthogonal polynomials relative to $\lambda p(c)q(c) + \langle{\tau},{p'(x)q'(x)}\rangle$, Comm. Korean Math. Soc. 12 (1997), 603–617.
  • R. Koekoek, P.A. Lesky and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Mono. Math., Springer-Verlag, Berlin, 2010.
  • A.B.J. Kuijlaars, A. Martinez-Finkelshtein and R. Orive, Orthogonality of Jacobi polynomials with general parameters, Electr. Trans. Numer. Anal. 19 (2005), 1–17.
  • K.H. Kwon and L.L. Littlejohn, The orthogonality of the Laguerre polynomials $\{L_n^{(-k)}(x)\}$ for a positive integer $k$, Ann. Num. Math. 2 (1995), 289–304.
  • S.G. Moreno and E.M. García-Caballero, Linear interpolation and Sobolev orthogonality, J. Appr. Th. 161 (2009), 35–48.
  • ––––, Non-standard orthogonality for the little $q$-Laguerre polynomials. Appl. Math. Lett. 22 (2009), 1745–1749.
  • ––––, Non-classical orthogonality relations for big and little $q$-Jacobi polynomials, J. Appr. Th. 162 (2010), 303–322.
  • ––––, New orthogonality relations for the continuous and the discrete $q$-ultraspherical polynomials, J. Math. Anal. Appl. 369 (2010), 386–399.
  • ––––, Non-classical orthogonality relations for continuous $q$-Jacobi polynomials. Taiwanese J. Math. 15 (2011), 1677–1690.
  • R.D. Morton and A.M. Krall, Distributional weight functions for orthogonal polynomials, SIAM J. Math. Anal. 9 (1978), 604–626.
  • T.E. Pérez and M.A. Piñar, On Sobolev orthogonality for the generalized Laguerre polynomials, J. Appr. Th. 86 (1996), 278–285.
  • G. Szegő, Orthogonal polynomials, AMS Colloq. Publ. 23, American Mathematical Society, Providence, RI, 1975.