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Askey–Wilson class orthogonal polynomials

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1: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for q -Hahn and AskeyWilson Classes
2: 18.28 Askey–Wilson Class
§18.28 AskeyWilson Class
§18.28(ii) AskeyWilson Polynomials
§18.28(viii) q -Racah Polynomials
Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).
3: 18.21 Hahn Class: Interrelations
See accompanying text
Figure 18.21.1: Askey scheme. … Magnify
4: 18.26 Wilson Class: Continued
Moreover, if one or more of the new parameters becomes zero, then the polynomial descends to a lower family in the Askey scheme.
5: 18.38 Mathematical Applications
See Zhedanov (1991), Granovskiĭ et al. (1992, §3), Koornwinder (2007a, §2) and Terwilliger (2011). … The Dunkl type operator is a q -difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric AskeyWilson polynomials, are linear combinations of the symmetric Laurent polynomial R n ( z ; a , b , c , d | q ) and the ‘anti-symmetric’ Laurent polynomial z 1 ( 1 a z ) ( 1 b z ) R n 1 ( z ; q a , q b , c , d | q ) , where R n ( z ) is given in (18.28.1_5). … Dunkl type operators and nonsymmetric polynomials have been associated with various other families in the Askey scheme and q -Askey scheme, in particular with Wilson polynomials, see Groenevelt (2007), and with Jacobi polynomials, see Koornwinder and Bouzeffour (2011, §7). …
6: Bibliography Z
  • D. Zagier (1989) The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.
  • Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
  • J. Zeng (1992) Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.
  • A. S. Zhedanov (1991) “Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.
  • A. Zhedanov (1998) On some classes of polynomials orthogonal on arcs of the unit circle connected with symmetric orthogonal polynomials on an interval. J. Approx. Theory 94 (1), pp. 73–106.
  • 7: 18.27 q -Hahn Class
    §18.27 q -Hahn Class
    The q -hypergeometric OP’s comprise the q -Hahn class (or q -linear lattice class) OP’s and the AskeyWilson class (or q -quadratic lattice class) OP’s (§18.28). … All these systems of OP’s have orthogonality properties of the form …
    From Big q -Jacobi to Jacobi
    Limit Relations
    8: 18.22 Hahn Class: Recurrence Relations and Differences
    §18.22 Hahn Class: Recurrence Relations and Differences
    §18.22(i) Recurrence Relations in n
    These polynomials satisfy (18.22.2) with p n ( x ) , A n , and C n as in Table 18.22.1. …
    §18.22(ii) Difference Equations in x
    §18.22(iii) x -Differences
    9: 18.19 Hahn Class: Definitions
    §18.19 Hahn Class: Definitions
  • 2.

    Wilson class (or quadratic lattice class). These are OP’s p n ( x ) = p n ( λ ( y ) ) ( p n ( x ) of degree n in x , λ ( y ) quadratic in y ) where the role of the differentiation operator is played by Δ y Δ y ( λ ( y ) ) or y y ( λ ( y ) ) or δ y δ y ( λ ( y ) ) . The Wilson class consists of two discrete and two continuous families.

  • Hahn, Krawtchouk, Meixner, and Charlier
    These polynomials are orthogonal on ( , ) , and are defined as follows. …A special case of (18.19.8) is w ( 1 / 2 ) ( x ; π / 2 ) = π cosh ( π x ) .
    10: 18.1 Notation
    Hahn Class OP’s
    Wilson Class OP’s
  • Wilson: W n ( x ; a , b , c , d ) .

  • AskeyWilson Class OP’s
  • AskeyWilson: p n ( x ; a , b , c , d | q ) .