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DLMF: §18.26 Wilson Class: Continued ‣ Askey Scheme ‣ Chapter 18 Orthogonal Polynomials
§18.26 Wilson Class: Continued
Contents
§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities§18.26(ii) Limit Relations§18.26(iii) Difference Relations§18.26(iv) Generating Functions§18.26(v) Asymptotic Approximations
§18.26(i) Representations as Generalized Hypergeometric Functions and Dualities
For the definition of generalized hypergeometric functions see §16.2 16.2.1 b q = − N a 1 = − n n = 0 , 1 , … , N k = n
18.26.1 W n ( y 2 ; a , b , c , d ) = ( a + b ) n ( a + c ) n ( a + d ) n F 3 4 ( − n , n + a + b + c + d − 1 , a + i y , a − i y a + b , a + c , a + d ; 1 ) .
18.26.2 S n ( y 2 ; a , b , c ) = ( a + b ) n ( a + c ) n F 2 3 ( − n , a + i y , a − i y a + b , a + c ; 1 ) .
18.26.3 R n ( y ( y + γ + δ + 1 ) ; α , β , γ , δ ) = F 3 4 ( − n , n + α + β + 1 , − y , y + γ + δ + 1 α + 1 , β + δ + 1 , γ + 1 ; 1 ) ,
α + 1 β + δ + 1 γ + 1 = − N n = 0 , 1 , … , N
18.26.4 R n ( y ( y + γ + δ + 1 ) ; γ , δ , N ) = F 2 3 ( − n , − y , y + γ + δ + 1 γ + 1 , − N ; 1 ) ,
n = 0 , 1 , … , N
Dualities
18.26.4_1 R n ( y ( y + γ + δ + 1 ) ; γ , δ , N ) = Q y ( n ; γ , δ , N ) ,
compare (18.21.1
18.26.4_2 R n ( y ( y + γ + δ + 1 ) ; α , β , γ , δ ) = R y ( n ( n + α + β + 1 ) ; γ , δ , α , β ) .
§18.26(ii) Limit Relations
Wilson →
18.26.5 lim d → ∞ W n ( x ; a , b , c , d ) ( a + d ) n = S n ( x ; a , b , c ) .
Wilson →
18.26.6 lim t → ∞ W n ( ( x + t ) 2 ; a − i t , b − i t , a ¯ + i t , b ¯ + i t ) ( − 2 t ) n n ! = p n ( x ; a , b , a ¯ , b ¯ ) .
Wilson →
18.26.7 lim t → ∞ W n ( 1 2 ( 1 − x ) t 2 ; 1 2 α + 1 2 , 1 2 α + 1 2 , 1 2 β + 1 2 + i t , 1 2 β + 1 2 − i t ) t 2 n n ! = P n ( α , β ) ( x ) .
Continuous Dual Hahn →
18.26.8 lim t → ∞ S n ( ( x − t ) 2 ; λ + i t , λ − i t , t cot ϕ ) / t n = n ! ( csc ϕ ) n P n ( λ ) ( x ; ϕ ) .
Racah →
18.26.9 lim β → ∞ R n ( x ; − N − 1 , β , γ , δ ) = R n ( x ; γ , δ , N ) .
Racah →
18.26.10 lim δ → ∞ R n ( x ( x + γ + δ + 1 ) ; α , β , − N − 1 , δ ) = Q n ( x ; α , β , N ) .
Dual Hahn →
18.26.11 lim t → ∞ R n ( x ( x + t + 1 ) ; p t , ( 1 − p ) t , N ) = K n ( x ; p , N ) .
Dual Hahn →
With
18.26.12 r ( x ; β , c , N ) = x ( x + β + c − 1 ( 1 − c ) N ) ,
18.26.13 lim N → ∞ R n ( r ( x ; β , c , N ) ; β − 1 , c − 1 ( 1 − c ) N , N ) = M n ( x ; β , c ) .
§18.26(iii) Difference Relations
For comments on the use of the forward-difference operator Δ x ∇ x δ x 18.2(ii)
For each family only the y n Koekoek et al. (2010 , Chapter 9) for further formulas.
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 ) .
18.26.15 δ y ( S n ( y 2 ; a , b , c ) ) / δ y ( y 2 ) = − n S n − 1 ( y 2 ; a + 1 2 , b + 1 2 , c + 1 2 ) .
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 ) .
18.26.17 Δ y ( R n ( y ( y + γ + δ + 1 ) ; γ , δ , N ) ) Δ y ( y ( y + γ + δ + 1 ) ) = − n ( γ + 1 ) N R n − 1 ( y ( y + γ + δ + 2 ) ; γ + 1 , δ , N − 1 ) .
§18.26(iv) Generating Functions
For the hypergeometric function F 1 2 15.1 15.2(i)
Wilson
18.26.18 F 1 2 ( a + i y , d + i y a + d ; z ) F 1 2 ( b − i y , c − i y b + c ; z ) = ∑ n = 0 ∞ W n ( y 2 ; a , b , c , d ) ( a + d ) n ( b + c ) n n ! z n ,
| z | < 1
Continuous Dual Hahn
18.26.19 ( 1 − z ) − c + i y F 1 2 ( a + i y , b + i y a + b ; z ) = ∑ n = 0 ∞ S n ( y 2 ; a , b , c ) ( a + b ) n n ! z n ,
| z | < 1
Racah
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 .
Dual Hahn
18.26.21 ( 1 − z ) y F 1 2 ( y − N , y + γ + 1 − δ − N ; z ) = ∑ n = 0 N ( γ + 1 ) n ( − N ) n ( − δ − N ) n n ! R n ( y ( y + γ + δ + 1 ) ; γ , δ , N ) z n .
§18.26(v) Asymptotic Approximations
For asymptotic expansions of Wilson polynomials of large degree see
Wilson (1991 ) , and for asymptotic approximations to their largest
zeros see Chen and Ismail (1998 ) .
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. Moreover, if one or more of the new parameters becomes zero, then the
polynomial descends to a lower family in the Askey scheme.
