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11: 18.4 Graphics
See accompanying text
Figure 18.4.1: Jacobi polynomials P n ( 1.5 , - 0.5 ) ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.2: Jacobi polynomials P n ( 1.25 , 0.75 ) ( x ) , n = 7 , 8 . …See also Askey (1990). Magnify
See accompanying text
Figure 18.4.4: Legendre polynomials P n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
See accompanying text
Figure 18.4.7: Monic Hermite polynomials h n ( x ) = 2 - n H n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
12: 18.37 Classical OP’s in Two or More Variables
In one variable they are essentially ultraspherical, Jacobi, continuous q -ultraspherical, or Askey–Wilson polynomials. …
13: 18.38 Mathematical Applications
The Askey--Gasper inequality …
14: Bibliography C
  • F. Calogero (1978) Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial L n α ( x )  as the index α  and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.
  • CAOP (website) Work Group of Computational Mathematics, University of Kassel, Germany.
  • 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.
  • 15: Bibliography I
  • A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna (1991) On polynomials orthogonal with respect to certain Sobolev inner products. J. Approx. Theory 65 (2), pp. 151–175.
  • M. E. H. Ismail and D. R. Masson (1994) q -Hermite polynomials, biorthogonal rational functions, and q -beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
  • M. E. H. Ismail (1986) Asymptotics of the Askey-Wilson and q -Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.
  • M. E. H. Ismail (2009) Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge.
  • 16: 18.17 Integrals
    The case x = 1 is a limit case of an integral for Jacobi polynomials; see Askey and Razban (1972). …
    17: 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.
  • 18: 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.
    19: 18.7 Interrelations and Limit Relations
    §18.7 Interrelations and Limit Relations
    Chebyshev, Ultraspherical, and Jacobi
    Legendre, Ultraspherical, and Jacobi
    See Figure 18.21.1 for the Askey schematic representation of most of these limits.
    20: 18.14 Inequalities
    Jacobi
    Ultraspherical
    Laguerre
    Hermite
    For extensions of (18.14.20) see Askey (1990) and Wong and Zhang (1994a, b). …