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Wilson class orthogonal polynomials

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1: 18.25 Wilson Class: Definitions
§18.25 Wilson Class: Definitions
Table 18.25.1: Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints.
p n ( x ) x = λ ( y ) Orthogonality range for y Constraints
§18.25(ii) Weights and Normalizations: Continuous Cases
18.25.15 h n = n ! ( N - n ) ! ( γ + δ + 2 ) N N ! ( γ + 1 ) n ( δ + 1 ) N - n .
Table 18.25.2: Wilson class OP’s: leading coefficients.
p n ( x ) k n
2: 18.26 Wilson Class: Continued
§18.26(i) Representations as Generalized Hypergeometric Functions
§18.26(ii) Limit Relations
§18.26(iii) Difference Relations
§18.26(iv) Generating Functions
§18.26(v) Asymptotic Approximations
3: 18.21 Hahn Class: Interrelations
§18.21(i) Dualities
See accompanying text
Figure 18.21.1: Askey scheme. … Magnify
4: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
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
§18.28(ii) Askey–Wilson Polynomials
For ω y and h n see Koekoek et al. (2010, Eq. (14.2.2)).
7: 18.27 q -Hahn Class
§18.27 q -Hahn Class
All these systems of OP’s have orthogonality properties of the form …
§18.27(ii) q -Hahn Polynomials
§18.27(iii) Big q -Jacobi Polynomials
Discrete q -Hermite II
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: Bibliography W
  • X.-S. Wang and R. Wong (2012) Asymptotics of orthogonal polynomials via recurrence relations. Anal. Appl. (Singap.) 10 (2), pp. 215–235.
  • G. N. Watson (1935a) Generating functions of class-numbers. Compositio Math. 1, pp. 39–68.
  • J. A. Wilson (1978) Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials. Ph.D. Thesis, University of Wisconsin, Madison, WI.
  • J. A. Wilson (1980) Some hypergeometric orthogonal polynomials. SIAM J. Math. Anal. 11 (4), pp. 690–701.
  • J. A. Wilson (1991) Asymptotics for the F 3 4 polynomials. J. Approx. Theory 66 (1), pp. 58–71.
  • 10: Bibliography C
  • CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
  • Y. Chen and M. E. H. Ismail (1998) Asymptotics of the largest zeros of some orthogonal polynomials. J. Phys. A 31 (25), pp. 5525–5544.
  • L. Chihara (1987) On the zeros of the Askey-Wilson polynomials, with applications to coding theory. SIAM J. Math. Anal. 18 (1), pp. 191–207.
  • T. S. Chihara (1978) An Introduction to Orthogonal Polynomials. Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York.
  • T. S. Chihara and M. E. H. Ismail (1993) Extremal measures for a system of orthogonal polynomials. Constr. Approx. 9, pp. 111–119.