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)],](/images/equations/PollaczekPolynomial/NumberedEquation3.svg) |
(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)](/images/equations/PollaczekPolynomial/NumberedEquation4.svg) |
(4)
|
for 
,
3, ... with
 |  |  |
(5)
|
 |  |  |
(6)
|
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),](/images/equations/PollaczekPolynomial/NumberedEquation6.svg) |
(8)
|
where 
is the Kronecker delta, for 
, 1, ..., with the weighting
function
![w(costheta;a,b)=e^((2theta-pi)h(theta)){cosh[pih(theta)]}^(-1).](/images/equations/PollaczekPolynomial/NumberedEquation7.svg) |
(9)
|
