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Pollaczek Polynomial


Let a>|b|, and write

 h(theta)=(acostheta+b)/(2sintheta).
(1)

Then define P_n(x;a,b) by the generating function

 f(x,w)=f(costheta,w)=sum_(n=0)^inftyP_n(x;a,b)w^n 
 =(1-we^(itheta))^(-1/2+ih(theta))(1-we^(itheta))^(-1/2-ih(theta)).
(2)

The generating function may also be written

 f(x,w)=(1-2xw+w^2)^(-1/2)exp[(ax+b)sum_(m=1)^infty(w^m)/mU_(m-1)(x)],
(3)

where U_m(x) is a Chebyshev polynomial of the second kind.

Pollaczek polynomials satisfy the recurrence relation

 nP_n(x;a,b)=[(2n-1+2a)x+2b]P_(n-1)(x;a,b)-(n-1)P_(n-2)(x;a,b)
(4)

for n=2, 3, ... with

P_0=1
(5)
P_1=(2a+1)x+2b.
(6)

In terms of the hypergeometric function _2F_1(a,b;c;x),

 P_n(costheta;a;b)=e^(intheta)_2F_1(-n,1/2+ih(theta);1;1-e^(-2itheta)).
(7)

They obey the orthogonality relation

 int_(-1)^1P_n(x;a,b)P_m(x;a,b)w(x;a,b)dx=[n+1/2(a+1)]^(-1)delta_(nm),
(8)

where delta_(nm) is the Kronecker delta, for n,m=0, 1, ..., with the weighting function

 w(costheta;a,b)=e^((2theta-pi)h(theta)){cosh[pih(theta)]}^(-1).
(9)

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References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., pp. 393-400, 1975.

Referenced on Wolfram|Alpha

Pollaczek Polynomial

Cite this as:

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

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