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Meixner-Pollaczek Polynomial

The hypergeometric orthogonal polynomials defined by

P_n^((lambda))(x;phi)=((2lambda)_n)/(n!)e^(inphi)_2F_1(-n,lambda+ix;2lambda;1-e^(-2iphi)),
(1)

where (x)_n is the Pochhammer symbol. The first few are given by

P_0^((lambda))(x;phi)=1
(2)
P_1^((lambda))(x;phi)=2(lambdacosphi+xsinphi)
(3)
P_2^((lambda))(x;phi)=x^2+lambda^2+(lambda^2+lambda-x^2)cos(2phi)+(1+2lambda)xsin(2phi).
(4)

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References

Koekoek, R. and Swarttouw, R. F. "Meixner-Pollaczek." §1.7 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. 37-38, 1998.

Referenced on Wolfram|Alpha

Meixner-Pollaczek Polynomial

Cite this as:

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

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