Pollaczek Polynomial
المؤلف:
Szegö, G.
المصدر:
Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc.,
الجزء والصفحة:
pp. 393-400
6-8-2019
2074
Pollaczek Polynomial
Let 
, and write
 |
(1)
|
Then define 
by the generating function
 |
(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)],](http://mathworld.wolfram.com/images/equations/PollaczekPolynomial/NumberedEquation3.gif) |
(3)
|
where 
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)](http://mathworld.wolfram.com/images/equations/PollaczekPolynomial/NumberedEquation4.gif) |
(4)
|
for 
, 3, ... with
In terms of the hypergeometric function 
,
 |
(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),](http://mathworld.wolfram.com/images/equations/PollaczekPolynomial/NumberedEquation6.gif) |
(8)
|
where 
is the Kronecker delta, for 
, 1, ..., with the weighting function
|
![w(costheta;a,b)=e^((2theta-pi)h(theta))<span style=]() {cosh[pih(theta)]}^(-1). " class="numberedequation" src="http://mathworld.wolfram.com/images/equations/PollaczekPolynomial/NumberedEquation7.gif" style="border:0px; height:17px; width:250px" />
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(9)
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