About the Project
NIST

Racah polynomials

AdvancedHelp

(0.001 seconds)

8 matching pages

1: 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
R n ( x ; α , β , γ , δ ) y ( y + γ + δ + 1 ) { 0 , 1 , , N } α + 1 or β + δ + 1 or γ + 1 = - N ; for further constraints see (18.25.1)
Further Constraints for Racah Polynomials
Table 18.25.2: Wilson class OP’s: leading coefficients.
p n ( x ) k n
R n ( x ; α , β , γ , δ ) ( n + α + β + 1 ) n ( α + 1 ) n ( β + δ + 1 ) n ( γ + 1 ) n
2: 18.26 Wilson Class: Continued
18.26.9 lim β R n ( x ; - N - 1 , β , γ , δ ) = R n ( x ; γ , δ , N ) .
18.26.10 lim δ R n ( x ( x + γ + δ + 1 ) ; α , β , - N - 1 , δ ) = Q n ( x ; α , β , N ) .
18.26.16 Δ y ( R n ( y ( y + γ + δ + 1 ) ; α , β , γ , δ ) ) Δ y ( y ( y + γ + δ + 1 ) ) = n ( n + α + β + 1 ) ( α + 1 ) ( β + δ + 1 ) ( γ + 1 ) R n - 1 ( y ( y + γ + δ + 2 ) ; α + 1 , β + 1 , γ + 1 , δ ) .
18.26.20 F 1 2 ( - y , - y + β - γ β + δ + 1 ; z ) F 1 2 ( y - N , y + γ + 1 - δ - N ; z ) = n = 0 N ( - N ) n ( γ + 1 ) n ( - δ - N ) n n ! R n ( y ( y + γ + δ + 1 ) ; - N - 1 , β , γ , δ ) z n .
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. …
3: 18.38 Mathematical Applications
Coding Theory
For applications of Krawtchouk polynomials K n ( x ; p , N ) and q -Racah polynomials R n ( x ; α , β , γ , δ | q ) to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).
4: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .
  • Racah: R n ( x ; α , β , γ , δ ) .

  • q -Racah: R n ( x ; α , β , γ , δ | 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 q -Racah polynomials form a system of OP’s { p n ( x ) } , n = 0 , 1 , 2 , , N , that are orthogonal with respect to a weight function on a sequence { q - y + c q y + 1 } , y = 0 , 1 , , N , with c a constant. Both the Askey–Wilson polynomials and the q -Racah polynomials can best be described as functions of z (resp. …
    §18.28(viii) q -Racah Polynomials
    18.28.19 R n ( x ) = R n ( x ; α , β , γ , δ | q ) = = 0 n q ( q - n , α β q n + 1 ; q ) ( α q , β δ q , γ q , q ; q ) j = 0 - 1 ( 1 - q j x + γ δ q 2 j + 1 ) = ϕ 3 4 ( q - n , α β q n + 1 , q - y , γ δ q y + 1 α q , β δ q , γ q ; q , q ) , α q , β δ q , or γ q = q - N ; n = 0 , 1 , , N .
    6: 16.4 Argument Unity
    One example of such a three-term relation is the recurrence relation (18.26.16) for Racah polynomials. … …
    7: 18.21 Hahn Class: Interrelations
    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
    8: Bibliography C
  • L. Chen, M. E. H. Ismail, and P. Simeonov (1999) Asymptotics of Racah coefficients and polynomials. J. Phys. A 32 (3), pp. 537–553.