Let , and write
(1)
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Then define by the generating function
(2)
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The generating function may also be written
(3)
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where is a Chebyshev polynomial of the second kind.
Pollaczek polynomials satisfy the recurrence relation
(4)
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for , 3, ... with
(5)
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(6)
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In terms of the hypergeometric function ,
(7)
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They obey the orthogonality relation
(8)
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where is the Kronecker delta, for , 1, ..., with the weighting function
(9)
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