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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
For ω y and h n see Koekoek et al. (2010, Eq. (14.2.2)).
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: 18.27 q -Hahn Class
§18.27 q -Hahn Class
The q -hypergeometric OP’s comprise the q -Hahn class OP’s and the AskeyWilson class OP’s (§18.28). … All these systems of OP’s have orthogonality properties of the form …
§18.27(ii) q -Hahn Polynomials
Discrete q -Hermite II
6: 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
7: 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 ) .

  • 8: 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.
  • 9: Bibliography M
  • I. G. Macdonald (1998) Symmetric Functions and Orthogonal Polynomials. University Lecture Series, Vol. 12, American Mathematical Society, Providence, RI.
  • I. G. Macdonald (2000) Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).
  • I. G. Macdonald (2003) Affine Hecke Algebras and Orthogonal Polynomials. Cambridge Tracts in Mathematics, Vol. 157, Cambridge University Press, Cambridge.
  • D. R. Masson (1991) Associated Wilson polynomials. Constr. Approx. 7 (4), pp. 521–534.
  • D. S. Moak (1981) The q -analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.