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continuous dual Hahn polynomials

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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
S n ( x ; a , b , c ) y 2 ( 0 , ) ( a , b , c ) > 0 ; nonreal parameters in conjugate pairs
18.25.6 p n ( x ) = S n ( x ; a 1 , a 2 , a 3 ) ,
Table 18.25.2 provides the leading coefficients k n 18.2(iii)) for the Wilson, continuous dual Hahn, Racah, and dual Hahn polynomials. …
2: 18.26 Wilson Class: Continued
18.26.5 lim d W n ( x ; a , b , c , d ) ( a + d ) n = S n ( x ; a , b , c ) .
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 ) .
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 ) .
18.26.19 ( 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 .
3: 18.1 Notation
( z 1 , , z k ; q ) = ( z 1 ; q ) ( z k ; q ) .