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Pollaczek Polynomial

Let $a>\vert b\vert$, and write

h(\theta)={a\cos\theta+b\over 2\sin\theta}.
\end{displaymath} (1)

Then define $P_n(x;a,b)$ by the Generating Function

f(x,w)=f(\cos\theta,w)=\sum_{n=0}^\infty P_n(x;a,b)w^n = (1-...
\end{displaymath} (2)

The Generating Function may also be written

f(x,w)=(1-2xw+w^2)^{-1/2}\mathop{\rm exp}\nolimits \left[{(ax+b)\sum_{m=1}^\infty {w^m\over m} U_{m-1}(x)}\right],
\end{displaymath} (3)

where $U_m(x)$ is a Chebyshev Polynomial of the Second Kind. They satisfy the Recurrence Relation

\end{displaymath} (4)

for $n=2$, 3, ...with
$\displaystyle P_0$ $\textstyle =$ $\displaystyle 1$ (5)
$\displaystyle P_1$ $\textstyle =$ $\displaystyle (2a+1)x+2b.$ (6)

In terms of the Hypergeometric Function ${}_2F_1(a,b;c;x)$,
P_n(\cos\theta; a; b)=e^{in\theta}{}_2F_1(-n, {\textstyle{1\over 2}}+ih(\theta); 1; 1-e^{-2i\theta}).
\end{displaymath} (7)

They obey the orthogonality relation

\int_{-1}^1 P_n(x; a,b)P_m(x; a,b)w(x; a,b)\,dx=[n+{\textstyle{1\over 2}}(a+1)]^{-1} \delta_{nm},
\end{displaymath} (8)

where $\delta_{nm}$ is the Kronecker Delta, for $n,m=0$, 1, ..., with the Weight Function
w(\cos\theta; a,b)=e^{(2\theta-\pi)h(\theta)} \{\cosh[\pi h(\theta)]\}^{-1}.
\end{displaymath} (9)


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

© 1996-9 Eric W. Weisstein