About the Project
NIST

Wilson polynomials

AdvancedHelp

(0.001 seconds)

1—10 of 24 matching pages

1: 18.26 Wilson Class: Continued
§18.26(i) Representations as Generalized Hypergeometric Functions
18.26.5 lim d W n ( x ; a , b , c , d ) ( a + d ) n = S n ( x ; a , b , c ) .
18.26.14 δ y ( W n ( y 2 ; a , b , c , d ) ) / δ y ( y 2 ) = - n ( n + a + b + c + d - 1 ) W n - 1 ( y 2 ; a + 1 2 , b + 1 2 , c + 1 2 , d + 1 2 ) .
§18.26(iv) Generating Functions
Koornwinder (2009) rescales and reparametrizes Racah polynomials and Wilson polynomials in such a way that they are continuous in their four parameters, provided that these parameters are nonnegative. …
2: 18.25 Wilson Class: Definitions
§18.25 Wilson Class: Definitions
Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials W n ( x ; a , b , c , d ) , continuous dual Hahn polynomials S n ( x ; a , b , c ) , Racah polynomials R n ( x ; α , β , γ , δ ) , and dual Hahn polynomials R n ( x ; γ , δ , N ) .
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
W n ( x ; a , b , c , d ) y 2 ( 0 , ) ( a , b , c , d ) > 0 ; nonreal parameters in conjugate pairs
18.25.3 p n ( x ) = W n ( x ; a 1 , a 2 , a 3 , a 4 ) ,
Table 18.25.2: Wilson class OP’s: leading coefficients.
p n ( x ) k n
W n ( x ; a , b , c , d ) ( - 1 ) n ( n + a + b + c + d - 1 ) n
3: 18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
§18.29 Asymptotic Approximations for q -Hahn and Askey–Wilson Classes
Ismail (1986) gives asymptotic expansions as n , with x and other parameters fixed, for continuous q -ultraspherical, big and little q -Jacobi, and Askey–Wilson polynomials. …For Askey–Wilson p n ( cos θ ; a , b , c , d | q ) the leading term is given by …
4: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
Wilson Class OP’s
  • Wilson: W n ( x ; a , b , c , d ) .

  • Askey–Wilson: p n ( x ; a , b , c , d | q ) .

  • 5: 18.28 Askey–Wilson Class
    The Askey–Wilson class OP’s comprise the four-parameter families of Askey–Wilson polynomials and of q -Racah polynomials, and cases of these families obtained by specialization of parameters. The Askey–Wilson polynomials form a system of OP’s { p n ( x ) } , n = 0 , 1 , 2 , , that are orthogonal with respect to a weight function on a bounded interval, possibly supplemented with discrete weights on a finite set. …Both the Askey–Wilson polynomials and the q -Racah polynomials can best be described as functions of z (resp. …
    §18.28(ii) Askey–Wilson Polynomials
    For ω y and h n see Koekoek et al. (2010, Eq. (14.2.2)).
    6: Tom H. Koornwinder
    Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC. …
    7: Richard A. Askey
     Wilson), introduced the Askey-Wilson polynomials. …
    8: 18.21 Hahn Class: Interrelations
    §18.21(i) Dualities
    See accompanying text
    Figure 18.21.1: Askey scheme. …It increases by one for each row ascended in the scheme, culminating with four free real parameters for the Wilson and Racah polynomials. … Magnify
    9: Bibliography I
  • M. E. H. Ismail, J. Letessier, G. Valent, and J. Wimp (1990) Two families of associated Wilson polynomials. Canad. J. Math. 42 (4), pp. 659–695.
  • M. E. H. Ismail (1986) Asymptotics of the Askey-Wilson and q -Jacobi polynomials. SIAM J. Math. Anal. 17 (6), pp. 1475–1482.
  • 10: 18.30 Associated OP’s
    For associated Askey–Wilson polynomials see Rahman (2001).