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Wilson Polynomial

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The orthogonal polynomials defined variously by

W_n(x^2;a,b,c,d)=(a+b)_n(a+c)_n(a+d)_n_4F_3(-n,a+b+c+d+n-1,a+ix,a-ix; a+b,a+c,a+d;1)
(1)

(Koekoek and Swarttouw 1998, p. 24) or

p_n(x;a,b,c,d)=W_n(-x^2;a,b,c,d)
(2)
=(a+b)_n(a+c)_n(a+d)_n_4F_3(-n,a+b+c+d+n-1,a-x,a+x; a+b,a+c,a+d;1)
(3)

(Koepf, p. 116, 1998).

The first few are

p_0(x;a,b,c,d)=1
(4)
p_1(x;a,b,c,d)=abc+abd+acd+bcd+(a+b+c+d)x^2.
(5)

The Wilson polynomials obey the identity

p_n(x;a,b,c,d)=p_n(x;b,a,c,d).
(6)

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References

Koekoek, R. and Swarttouw, R. F. "Wilson." §1.1 in The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 24-26, 1998.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, p. 116, 1998.Wilson, J. A. "Some Hypergeometric Orthogonal Polynomials." SIAM J. Math. Anal. 11, 690-701, 1980.

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Wilson Polynomial

Cite this as:

Weisstein, Eric W. "Wilson Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WilsonPolynomial.html

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