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Askey scheme for orthogonal polynomials

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1: 18.21 Hahn Class: Interrelations
See accompanying text
Figure 18.21.1: Askey scheme. … Magnify
2: Richard A. Askey
Published in 1985 in the Memoirs of the American Mathematical Society, it also introduced the directed graph of hypergeometric orthogonal polynomials commonly known as the Askey scheme. …
3: Bibliography T
  • N. M. Temme and J. L. López (2001) The Askey scheme for hypergeometric orthogonal polynomials viewed from asymptotic analysis. J. Comput. Appl. Math. 133 (1-2), pp. 623–633.
  • 4: 18.19 Hahn Class: Definitions
    §18.19 Hahn Class: Definitions
    The Askey scheme extends the three families of classical OP’s (Jacobi, Laguerre and Hermite) with eight further families of OP’s for which the role of the differentiation operator d d x in the case of the classical OP’s is played by a suitable difference operator. …In addition to the limit relations in §18.7(iii) there are limit relations involving the further families in the Askey scheme, see §§18.21(ii) and 18.26(ii). The Askey scheme, depicted in Figure 18.21.1, gives a graphical representation of these limits. …
    Hahn, Krawtchouk, Meixner, and Charlier
    5: Bibliography K
  • R. Koekoek and R. F. Swarttouw (1998) The Askey-scheme of hypergeometric orthogonal polynomials and its q -analogue. Technical report Technical Report 98-17, Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics.
  • 6: 18.27 q -Hahn Class
    §18.27(i) Introduction
    Together they form the q -Askey scheme. … All these systems of OP’s have orthogonality properties of the form …
    §18.27(ii) q -Hahn Polynomials
    Limit Relations
    7: 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.
    8: 18.38 Mathematical Applications
    The Askey–Gasper inequality …also the case β = 0 of (18.14.26), was used in de Branges’ proof of the long-standing Bieberbach conjecture concerning univalent functions on the unit disk in the complex plane. …
    Group Representations
    Similar algebras can be associated with all families of OP’s in the q -Askey scheme and the Askey scheme. … 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). …
    9: Bibliography C
  • CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
  • 10: Bibliography S
  • H. E. Salzer (1955) Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.
  • B. Simon (2005a) Orthogonal Polynomials on the Unit Circle. Part 1: Classical Theory. American Mathematical Society Colloquium Publications, Vol. 54, American Mathematical Society, Providence, RI.
  • R. F. Swarttouw (1997) A computer implementation of the Askey-Wilson scheme. Technical Report 13 Vrije Universteit Amsterdam.
  • G. Szegö (1950) On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, pp. 731–737.
  • G. Szegő (1967) Orthogonal Polynomials. 3rd edition, American Mathematical Society, New York.