18 Orthogonal PolynomialsOther Orthogonal Polynomials18.34 Bessel Polynomials18.36 Miscellaneous Polynomials

- §18.35(i) Definition and Hypergeometric Representation
- §18.35(ii) Orthogonality
- §18.35(iii) Other Properties

There are 3 types of Pollaczek polynomials:

18.35.0_5 | ${P}_{n}^{(\frac{1}{2})}(x;a,b),$ | ||

${P}_{n}^{(\lambda )}(x;a,b)$ | $={P}_{n}^{(\lambda )}(x;a,b,0),$ | ||

${P}_{n}^{(\lambda )}(x;a,b,c).$ | |||

Thus type 3 with $c=0$ reduces to type 2, and type 3 with $c=0$ and $\lambda =\frac{1}{2}$ reduces to type 1, also in subsequent formulas. The three types of Pollaczek polynomials were successively introduced in Pollaczek (1949a, b, 1950), see also Erdélyi et al. (1953b, p.219) and, for type 1 and 2, Szegö (1950) and Askey (1982b). The type 2 polynomials reduce for $a=b=0$ to ultraspherical polynomials, see (18.35.8).

The Pollaczek polynomials of type 3 are defined by the recurrence relation (in first form (18.2.8))

18.35.1 | ${P}_{-1}^{(\lambda )}(x;a,b,c)$ | $=0,$ | ||

${P}_{0}^{(\lambda )}(x;a,b,c)$ | $=1,$ | |||

18.35.2 | ${P}_{n+1}^{(\lambda )}(x;a,b,c)$ | $={\displaystyle \frac{2(n+c+\lambda +a)x+2b}{n+c+1}}{P}_{n}^{(\lambda )}(x;a,b,c)-{\displaystyle \frac{n+c+2\lambda -1}{n+c+1}}{P}_{n-1}^{(\lambda )}(x;a,b,c),$ | ||

$n=0,1,\mathrm{\dots}$, | ||||

or, equivalently in second form (18.2.10),

18.35.2_1 | $$x{P}_{n}^{(\lambda )}(x;a,b,c)=\begin{array}{l}\frac{n+c+1}{2(n+c+\lambda +a)}{P}_{n+1}^{(\lambda )}(x;a,b,c)-\frac{b}{n+c+\lambda +a}{P}_{n}^{(\lambda )}(x;a,b,c)\\ \phantom{\rule{2em}{0ex}}+\frac{n+c+2\lambda -1}{2(n+c+\lambda +a)}{P}_{n-1}^{(\lambda )}(x;a,b,c),\end{array}$$ | ||

$n=0,1,\mathrm{\dots}$. | |||

For the monic polynomials

18.35.2_2 | $${Q}_{n}^{(\lambda )}(x;a,b,c)=\frac{{\left(c+1\right)}_{n}}{{2}^{n}{\left(c+\lambda +a\right)}_{n}}{P}_{n}^{(\lambda )}(x;a,b,c)$$ | ||

the recurrence relation of form (18.2.11_5) becomes

18.35.2_3 | ${Q}_{-1}^{(\lambda )}(x;a,b,c)$ | $=0,$ | ||

${Q}_{0}^{(\lambda )}(x;a,b,c)$ | $=1,$ | |||

18.35.2_4 | $x{Q}_{n}^{(\lambda )}(x;a,b,c)$ | $=\begin{array}{l}{Q}_{n+1}^{(\lambda )}(x;a,b,c)-{\displaystyle \frac{b}{n+c+\lambda +a}}{Q}_{n}^{(\lambda )}(x;a,b,c)\\ \phantom{\rule{2em}{0ex}}+{\displaystyle \frac{(n+c)(n+c+2\lambda -1)}{4(n+c+\lambda +a-1)(n+c+\lambda +a)}}{Q}_{n-1}^{(\lambda )}(x;a,b,c),\end{array}$ | ||

$n=0,1,\mathrm{\dots}$. | ||||

There is the symmetry

18.35.2_5 | $${P}_{n}^{(\lambda )}(-x;a,b,c)={(-1)}^{n}{P}_{n}^{(\lambda )}(x;a,-b,c).$$ | ||

As in the coefficients of the above recurrence relations $n$ and $c$ only
occur in the form $n+c$, the type 3 Pollaczek polynomials
may also be called the
*associated type 2 Pollaczek polynomials*
by using the terminology of §18.30.

For type 2, with notation

18.35.3 | $${\tau}_{a,b}(\theta )=\frac{a\mathrm{cos}\theta +b}{\mathrm{sin}\theta},$$ | ||

$$, | |||

we have the explicit representations

18.35.4 | $$\begin{array}{ll}{P}_{n}^{(\lambda )}(\mathrm{cos}\theta ;a,b)& =\frac{{\left(\lambda -\mathrm{i}{\tau}_{a,b}(\theta )\right)}_{n}}{n!}{\mathrm{e}}^{\mathrm{i}n\theta}F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}{\tau}_{a,b}(\theta )}{-n-\lambda +1+\mathrm{i}{\tau}_{a,b}(\theta )};{\mathrm{e}}^{-2\mathrm{i}\theta})\\ & =\sum _{\mathrm{\ell}=0}^{n}\frac{{\left(\lambda +\mathrm{i}{\tau}_{a,b}(\theta )\right)}_{\mathrm{\ell}}}{\mathrm{\ell}!}\frac{{\left(\lambda -\mathrm{i}{\tau}_{a,b}(\theta )\right)}_{n-\mathrm{\ell}}}{(n-\mathrm{\ell})!}{\mathrm{e}}^{\mathrm{i}(n-2\mathrm{\ell})\theta},\end{array}$$ | ||

18.35.4_5 | $${P}_{n}^{(\lambda )}(\mathrm{cos}\theta ;a,b)=\frac{{\left(2\lambda \right)}_{n}}{n!}{\mathrm{e}}^{\mathrm{i}n\theta}F(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}{\tau}_{a,b}(\theta )}{2\lambda};1-{\mathrm{e}}^{-2\mathrm{i}\theta}).$$ | ||

For type 1 take $\lambda =\frac{1}{2}$ and for Gauss’ hypergeometric function $F$ see (15.2.1).

First consider type 2.

18.35.5 | $${\int}_{-1}^{1}{P}_{n}^{(\lambda )}(x;a,b){P}_{m}^{(\lambda )}(x;a,b){w}^{(\lambda )}(x;a,b)dx=\frac{\mathrm{\Gamma}\left(2\lambda +n\right)}{n!(\lambda +a+n)}{\delta}_{n,m},$$ | ||

$a\ge b\ge -a$, $\lambda >0$, | |||

where

18.35.6 | $${w}^{(\lambda )}(\mathrm{cos}\theta ;a,b)={\pi}^{-1}{\mathrm{e}}^{(2\theta -\pi ){\tau}_{a,b}(\theta )}{\left(2\mathrm{sin}\theta \right)}^{2\lambda -1}{\left|\mathrm{\Gamma}\left(\lambda +\mathrm{i}{\tau}_{a,b}(\theta )\right)\right|}^{2},$$ | ||

$$. | |||

Note that

18.35.6_1 | $$\begin{array}{l}\mathrm{ln}\left({w}^{(\lambda )}(\mathrm{cos}\theta ;a,b)\right)\\ \phantom{\rule{2em}{0ex}}=\{\begin{array}{cc}-2\pi (a+b){\theta}^{-1}+(2\lambda -1)\mathrm{ln}\left(a+b\right)+\lambda \mathrm{ln}4+2(a+b)+O\left(\theta \right),\hfill & \theta \to 0+,\hfill \\ 2\pi (b-a){\left(\pi -\theta \right)}^{-1}+(2\lambda -1)\mathrm{ln}\left(a-b\right)+\lambda \mathrm{ln}4+2(a-b)+O\left(\pi -\theta \right),\hfill & \theta \to \pi -,\hfill \end{array}\end{array}$$ | ||

indicating the presence of essential singularities. Hence, only in the case $a=b=0$ does $\mathrm{ln}\left({w}^{(\lambda )}(x;a,b)\right)$ satisfy the condition (18.2.39) for the Szegő class $\mathcal{G}$.

More generally, the ${P}_{n}^{(\lambda )}(x;a,b)$ are OP’s if and only if one of the following three conditions holds (in case (iii) work with the monic polynomials (18.35.2_2)).

18.35.6_2 | $$ | ||

Then

18.35.6_3 | $$\begin{array}{l}\begin{array}{l}{\int}_{-1}^{1}{P}_{n}^{(\lambda )}(x;a,b){P}_{m}^{(\lambda )}(x;a,b){w}^{(\lambda )}(x;a,b)dx\\ \phantom{\rule{2em}{0ex}}+\sum _{\zeta \in D}{P}_{n}^{(\lambda )}(\zeta ;a,b){P}_{m}^{(\lambda )}(\zeta ;a,b){w}_{\zeta}^{(\lambda )}(a,b)\end{array}\\ \phantom{\rule{2em}{0ex}}=\frac{\mathrm{\Gamma}\left(2\lambda +n\right)}{n!(\lambda +a+n)}{\delta}_{n,m},\end{array}$$ | ||

where, depending on $a,b,\lambda $, $D$ is a discrete subset of $\mathbb{R}$ and the ${w}_{\zeta}^{(\lambda )}(a,b)$ are certain weights. See Ismail (2009, §5.5). In particular, if $a>b>-a$ and condition (ii) of (18.35.6_2) holds then $|D|=2$ (see Ismail (2009, Theorem 5.5.1)). Also, if $b>a\ge -b$, $\lambda +a>0$ then

18.35.6_4 | $D$ | $={\left\{{x}_{k}={\displaystyle \frac{(\lambda +k)\mathrm{\Delta}-ab}{{a}^{2}-{\left(\lambda +k\right)}^{2}}}\right\}}_{k=0}^{\mathrm{\infty}},$ | ||

${w}_{{x}_{k}}^{(\lambda )}(a,b)$ | $={\displaystyle \frac{{\rho}^{2k-1}{\left(1-{\rho}^{2}\right)}^{2\lambda +1}\mathrm{\Gamma}\left(2\lambda +k\right)}{2\mathrm{\Delta}k!}},$ | |||

$\mathrm{\Delta}$ | $=\sqrt{{\left(\lambda +k\right)}^{2}+{b}^{2}-{a}^{2}},$ | |||

$\rho $ | $={\displaystyle \frac{\mathrm{\Delta}-b}{\lambda +k-a}},$ | |||

and similarly if $-b\ge a>b$, $\lambda +a>0$ by application of (18.35.2_5).

For type 3 orthogonality (18.35.5) generalizes to

18.35.6_5 | $${\int}_{-1}^{1}{P}_{n}^{(\lambda )}(x;a,b,c){P}_{m}^{(\lambda )}(x;a,b,c){w}^{(\lambda )}(x;a,b,c)dx=\frac{\mathrm{\Gamma}\left(c+1\right)\mathrm{\Gamma}\left(2\lambda +c+n\right)}{{\left(c+1\right)}_{n}(\lambda +a+c+n)}{\delta}_{n,m},$$ | ||

where

18.35.6_6 | $${w}^{(\lambda )}(\mathrm{cos}\theta ;a,b,c)=\frac{{\mathrm{e}}^{(2\theta -\pi ){\tau}_{a,b}(\theta )}{\left(2\mathrm{sin}\theta \right)}^{2\lambda -1}{\left|\mathrm{\Gamma}\left(c+\lambda +\mathrm{i}{\tau}_{a,b}(\theta )\right)\right|}^{2}}{\pi {\left|F(\genfrac{}{}{0pt}{}{1-\lambda +\mathrm{i}{\tau}_{a,b}(\theta ),c}{c+\lambda +\mathrm{i}{\tau}_{a,b}(\theta )};{\mathrm{e}}^{2\mathrm{i}\theta})\right|}^{2}},$$ | ||

with two possible constraints: $a>b>-a$, $2\lambda +c>0$, $c\ge 0$, or $a>b>-a$, $2\lambda +c\ge 1$, $c>-1$. For Gauss’ hypergeometric function $F$ see (15.2.1).

18.35.7 | $${(1-z{\mathrm{e}}^{\mathrm{i}\theta})}^{-\lambda +\mathrm{i}{\tau}_{a,b}(\theta )}{(1-z{\mathrm{e}}^{-\mathrm{i}\theta})}^{-\lambda -\mathrm{i}{\tau}_{a,b}(\theta )}=\sum _{n=0}^{\mathrm{\infty}}{P}_{n}^{(\lambda )}(\mathrm{cos}\theta ;a,b){z}^{n},$$ | ||

$$, $$. | |||

18.35.8 | $${P}_{n}^{(\lambda )}(x;0,0)={C}_{n}^{(\lambda )}\left(x\right),$$ | ||

18.35.9 | ${P}_{n}^{(\lambda )}(x;\varphi )$ | $={P}_{n}^{(\lambda )}(\mathrm{cos}\varphi ;0,x\mathrm{sin}\varphi ),$ | ||

${P}_{n}^{(\lambda )}(\mathrm{cos}\theta ;a,b)$ | $={P}_{n}^{(\lambda )}({\tau}_{a,b}(\theta );\theta ),$ | |||

18.35.10 | $${\mathcal{P}}_{n}^{\lambda}(x;\varphi ,c)={P}_{n}^{(\lambda )}(\mathrm{cos}\varphi ;0,x\mathrm{sin}\varphi ,c).$$ | ||

For the ultraspherical polynomials ${C}_{n}^{(\lambda )}\left(x\right)$, the Meixner–Pollaczek polynomials ${P}_{n}^{(\lambda )}(x;\varphi )$ and the associated Meixner–Pollaczek polynomials ${\mathcal{P}}_{n}^{\lambda}(x;\varphi ,c)$ see §§18.3, 18.19 and 18.30(v), respectively.

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