§18.35(i) Definition and Hypergeometric Representation

 18.35.1 $\displaystyle\mathop{P^{(\lambda)}_{-1}\/}\nolimits\!\left(x;a,b\right)$ $\displaystyle=0,$ $\displaystyle\mathop{P^{(\lambda)}_{0}\/}\nolimits\!\left(x;a,b\right)$ $\displaystyle=1,$

and

 18.35.2 $(n+1)\mathop{P^{(\lambda)}_{n+1}\/}\nolimits\!\left(x;a,b\right)={2((n+\lambda% +a)x+b)}\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(x;a,b\right)-{(n+2\lambda% -1)}\mathop{P^{(\lambda)}_{n-1}\/}\nolimits\!\left(x;a,b\right),$ $n=0,1,\dots$.

Next, let

 18.35.3 $\tau_{a,b}(\theta)=\frac{a\mathop{\cos\/}\nolimits\theta+b}{\mathop{\sin\/}% \nolimits\theta},$ $0<\theta<\pi$. Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $\mathop{\sin\/}\nolimits z$: sine function and $\tau_{a,b}(\theta)$ Permalink: http://dlmf.nist.gov/18.35.E3 Encodings: TeX, pMML, png

Then

 18.35.4 $\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta;a,% b\right)=\frac{\left(\lambda-i\tau_{a,b}(\theta)\right)_{n}}{n!}e^{in\theta}\*% \mathop{{{}_{2}F_{1}}\/}\nolimits\!\left({-n,\lambda+i\tau_{a,b}(\theta)\atop-% n-\lambda+1+i\tau_{a,b}(\theta)};e^{-2i\theta}\right)=\sum_{\ell=0}^{n}\frac{% \left(\lambda+i\tau_{a,b}(\theta)\right)_{\ell}}{\ell!}\frac{\left(\lambda-i% \tau_{a,b}(\theta)\right)_{n-\ell}}{(n-\ell)!}e^{i(n-2\ell)\theta}.$

For the hypergeometric function $\mathop{{{}_{2}F_{1}}\/}\nolimits$ see §§15.1, 15.2(i).

§18.35(ii) Orthogonality

 18.35.5 ${\int_{-1}^{1}\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(x;a,b\right)\mathop% {P^{(\lambda)}_{m}\/}\nolimits\!\left(x;a,b\right)w^{(\lambda)}(x;a,b)dx=0},$ $n\neq m$,

where

 18.35.6 $w^{(\lambda)}(\mathop{\cos\/}\nolimits\theta;a,b)=\pi^{-1}\*2^{2\lambda-1}\*e^% {(2\theta-\pi)\*\tau_{a,b}(\theta)}\*(\mathop{\sin\/}\nolimits\theta)^{2% \lambda-1}\*\left|\mathop{\Gamma\/}\nolimits\!\left(\lambda+i\tau_{a,b}(\theta% )\right)\right|^{2},$ $a\geq b\geq-a$, $\lambda>-\frac{1}{2}$, $0<\theta<\pi$.

§18.35(iii) Other Properties

 18.35.7 $(1-ze^{i\theta})^{-\lambda+i\tau_{a,b}(\theta)}(1-ze^{-i\theta})^{-\lambda-i% \tau_{a,b}(\theta)}=\sum_{n=0}^{\infty}\mathop{P^{(\lambda)}_{n}\/}\nolimits\!% \left(\mathop{\cos\/}\nolimits\theta;a,b\right)z^{n},$ $|z|<1$, $0<\theta<\pi$.
 18.35.8 $\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(x;0,0\right)=\mathop{C^{(\lambda)% }_{n}\/}\nolimits\!\left(x\right),$
 18.35.9 $\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\phi;0,x% \mathop{\sin\/}\nolimits\phi\right)=\mathop{P^{(\lambda)}_{n}\/}\nolimits\!% \left(x;\phi\right).$

For the polynomials $\mathop{C^{(\lambda)}_{n}\/}\nolimits\!\left(x\right)$ and $\mathop{P^{(\lambda)}_{n}\/}\nolimits\!\left(x;\phi\right)$ see §§18.3 and 18.19, respectively.

See Bo and Wong (1996) for an asymptotic expansion of $\mathop{P^{(\frac{1}{2})}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits(n^{-% \frac{1}{2}}\theta);a,b\right)$ as $n\to\infty$, with $a$ and $b$ fixed. This expansion is in terms of the Airy function $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ and its derivative (§9.2), and is uniform in any compact $\theta$-interval in $(0,\infty)$. Also included is an asymptotic approximation for the zeros of $\mathop{P^{(\frac{1}{2})}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits(n^{-% \frac{1}{2}}\theta);a,b\right)$.