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Wilson class orthogonal polynomials

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1: 18.25 Wilson Class: Definitions
§18.25 Wilson Class: Definitions
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Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
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OP p n ⁡ ( x ) x = λ ⁢ ( y ) Orthogonality range for y Constraints
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§18.25(ii) Weights and Standardizations: Continuous Cases
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18.25.15 h n = n ! ⁢ ( N n ) ! ⁢ ( γ + δ + 2 ) N N ! ⁢ ( γ + 1 ) n ⁢ ( δ + 1 ) N n .
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Table 18.25.2: Wilson class OP’s: leading coefficients.
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p n ⁡ ( x ) k n
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2: 18.26 Wilson Class: Continued
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§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities
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§18.26(ii) Limit Relations
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§18.26(iii) Difference Relations
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§18.26(iv) Generating Functions
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§18.26(v) Asymptotic Approximations
3: 18.21 Hahn Class: Interrelations
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§18.21(i) Dualities
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►See accompanying text►
Figure 18.21.1: Askey scheme. … Magnify
4: 18.1 Notation
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( z 1 , , z k ; q ) = ( z 1 ; q ) ⁢ ⋯ ⁢ ( z k ; q ) .
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Wilson Class OP’s
5: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
6: 18.28 Askey–Wilson Class
§18.28 Askey–Wilson Class
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§18.28(ii) Askey–Wilson Polynomials
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§18.28(x) Limit Relations
►Genest et al. (2016) showed that these polynomials coincide with the nonsymmetric Wilson polynomials in Groenevelt (2007).
7: 18.38 Mathematical Applications
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8: Bibliography Z
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  • Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
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  • J. Zeng (1992) Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.
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  • A. S. Zhedanov (1991) “Hidden symmetry” of Askey-Wilson polynomials. Theoret. and Math. Phys. 89 (2), pp. 1146–1157.
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  • 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.
  • 9: 18.27 q -Hahn Class
    §18.27 q -Hahn Class
    ►All these systems of OP’s have orthogonality properties of the form … ►
    From Big q -Jacobi to Jacobi
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    From Big q -Jacobi to Little q -Jacobi
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    Limit Relations
    10: 18.22 Hahn Class: Recurrence Relations and Differences
    §18.22 Hahn Class: Recurrence Relations and Differences
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    §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
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    §18.22(iii) x -Differences