Pollaczek Polynomial

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

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

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

$$f(x,w)=f(\cos\theta,w)=\sum_{n=0}^\infty P_n(x;a,b)w^n = (1-... ...i\theta})^{-1/2+ih(\theta)}(1-we^{i\theta})^{-1/2-ih(\theta)}.$$ (2)

The Generating Function may also be written

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

where $U_m(x)$ is a Chebyshev Polynomial of the Second Kind. They 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 ${}_2F_1(a,b;c;x)$,

$$P_n(\cos\theta; a; b)=e^{in\theta}{}_2F_1(-n, {\textstyle{1\over 2}}+ih(\theta); 1; 1-e^{-2i\theta}).$$ (7)

They obey the orthogonality relation

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

where $\delta_{nm}$ is the Kronecker Delta, for $n,m=0$, 1, ..., with the Weight Function

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

References

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