Wilson Polynomial
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)
(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.Referenced on Wolfram|Alpha
Wilson PolynomialCite this as:
Weisstein, Eric W. "Wilson Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/WilsonPolynomial.html