# §18.35 Pollaczek Polynomials

## §18.35(i) Definition and Hypergeometric Representation

There are 3 types of Pollaczek polynomials:

 18.35.0_5 $P^{(\frac{1}{2})}_{n}\left(x;a,b\right),$ $\displaystyle P^{(\lambda)}_{n}\left(x;a,b\right)$ $\displaystyle=P^{{(\lambda)}}_{n}\left(x;a,b,0\right),$ $P^{{(\lambda)}}_{n}\left(x;a,b,c\right).$ ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.35(i), Erratum (V1.2.0) §18.35 Permalink: http://dlmf.nist.gov/18.35.E0_5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(i), §18.35 and Ch.18

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 $\displaystyle P^{{(\lambda)}}_{-1}\left(x;a,b,c\right)$ $\displaystyle=0,$ $\displaystyle P^{{(\lambda)}}_{0}\left(x;a,b,c\right)$ $\displaystyle=1,$ ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial and $x$: real variable Source: Erdélyi et al. (1953b, 10.21(12)) Referenced by: (18.35.2_5), Erratum (V1.2.0) for Equation (18.35.1) Permalink: http://dlmf.nist.gov/18.35.E1 Encodings: TeX, TeX, pMML, pMML, png, png Update (effective with 1.2.0): These equations which were previously given for Pollaczek polynomials of type 2 has been updated for Pollaczek polynomials of type 3. See also: Annotations for §18.35(i), §18.35 and Ch.18 18.35.2 $\displaystyle P^{{(\lambda)}}_{n+1}\left(x;a,b,c\right)$ $\displaystyle=\frac{2(n+c+\lambda+a)x+2b}{n+c+1}\,P^{{(\lambda)}}_{n}\left(x;a% ,b,c\right)-\frac{n+c+2\lambda-1}{n+c+1}\,P^{{(\lambda)}}_{n-1}\left(x;a,b,c% \right),$ $n=0,1,\dots$, ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable Source: Erdélyi et al. (1953b, 10.21(13)) Referenced by: (18.35.2_5), §18.39(iv), Erratum (V1.2.0) for Equation (18.35.2) Permalink: http://dlmf.nist.gov/18.35.E2 Encodings: TeX, pMML, png See also: Annotations for §18.35(i), §18.35 and Ch.18

or, equivalently in second form (18.2.10),

 18.35.2_1 $xP^{{(\lambda)}}_{n}\left(x;a,b,c\right)=\frac{n+c+1}{2(n+c+\lambda+a)}\,P^{{(% \lambda)}}_{n+1}\left(x;a,b,c\right)-\frac{b}{n+c+\lambda+a}\,P^{{(\lambda)}}_% {n}\left(x;a,b,c\right)+\frac{n+c+2\lambda-1}{2(n+c+\lambda+a)}\,P^{{(\lambda)% }}_{n-1}\left(x;a,b,c\right),$ $n=0,1,\dots$. ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.35(i), Erratum (V1.2.0) §18.35 Permalink: http://dlmf.nist.gov/18.35.E2_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(i), §18.35 and Ch.18

For the monic polynomials

 18.35.2_2 ${Q}^{{(\lambda)}}_{n}\left(x;a,b,c\right)=\frac{{\left(c+1\right)_{n}}}{2^{n}{% \left(c+\lambda+a\right)_{n}}}\,P^{{(\lambda)}}_{n}\left(x;a,b,c\right)$ ⓘ Defines: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer and $x$: real variable Referenced by: §18.35(ii) Permalink: http://dlmf.nist.gov/18.35.E2_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(i), §18.35 and Ch.18

the recurrence relation of form (18.2.11_5) becomes

 18.35.2_3 $\displaystyle{Q}^{{(\lambda)}}_{-1}\left(x;a,b,c\right)$ $\displaystyle=0,$ $\displaystyle{Q}^{{(\lambda)}}_{0}\left(x;a,b,c\right)$ $\displaystyle=1,$ ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial and $x$: real variable Permalink: http://dlmf.nist.gov/18.35.E2_3 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(i), §18.35 and Ch.18 18.35.2_4 $\displaystyle x{Q}^{{(\lambda)}}_{n}\left(x;a,b,c\right)$ $\displaystyle={Q}^{{(\lambda)}}_{n+1}\left(x;a,b,c\right)-\frac{b}{n+c+\lambda% +a}{Q}^{{(\lambda)}}_{n}\left(x;a,b,c\right)+\frac{(n+c)(n+c+2\lambda-1)}{4(n+% c+\lambda+a-1)(n+c+\lambda+a)}{Q}^{{(\lambda)}}_{n-1}\left(x;a,b,c\right),$ $n=0,1,\dots$. ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: (18.35.6_2) Permalink: http://dlmf.nist.gov/18.35.E2_4 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(i), §18.35 and Ch.18

There is the symmetry

 18.35.2_5 $P^{{(\lambda)}}_{n}\left(-x;a,b,c\right)=(-1)^{n}P^{{(\lambda)}}_{n}\left(x;a,% -b,c\right).$ ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable Proof sketch: Use (18.35.1), (18.35.2). Referenced by: §18.35(i), §18.35(ii), Erratum (V1.2.0) §18.35 Permalink: http://dlmf.nist.gov/18.35.E2_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(i), §18.35 and Ch.18

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 $\tau_{a,b}(\theta)=\frac{a\cos\theta+b}{\sin\theta},$ $0<\theta<\pi$, ⓘ Defines: $\tau_{a,b}(\theta)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function and $\sin\NVar{z}$: sine function Referenced by: §18.38(ii) Permalink: http://dlmf.nist.gov/18.35.E3 Encodings: TeX, pMML, png See also: Annotations for §18.35(i), §18.35 and Ch.18

we have the explicit representations

 18.35.4 $P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=\frac{{\left(\lambda-\mathrm{i}% \tau_{a,b}(\theta)\right)_{n}}}{n!}\,{\mathrm{e}}^{\mathrm{i}n\theta}\*F\left(% {-n,\lambda+\mathrm{i}\tau_{a,b}(\theta)\atop-n-\lambda+1+\mathrm{i}\tau_{a,b}% (\theta)};{\mathrm{e}}^{-2\mathrm{i}\theta}\right)=\sum_{\ell=0}^{n}\frac{{% \left(\lambda+\mathrm{i}\tau_{a,b}(\theta)\right)_{\ell}}}{\ell!}\,\frac{{% \left(\lambda-\mathrm{i}\tau_{a,b}(\theta)\right)_{n-\ell}}}{(n-\ell)!}\,{% \mathrm{e}}^{\mathrm{i}(n-2\ell)\theta},$ ⓘ Defines: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial Symbols: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$: $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\ell$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $\tau_{a,b}(\theta)$ Proved: Ismail (2009, (5.4.10)); In the cited formula, in the lower parameter of the hypergeometric function, $-\lambda$ should be replaced by $-\lambda+1$.(proved) Referenced by: (18.35.4_5) Permalink: http://dlmf.nist.gov/18.35.E4 Encodings: TeX, pMML, png See also: Annotations for §18.35(i), §18.35 and Ch.18
 18.35.4_5 $P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=\frac{{\left(2\lambda\right)_{n}}% }{n!}\,{\mathrm{e}}^{\mathrm{i}n\theta}\*F\left({-n,\lambda+\mathrm{i}\tau_{a,% b}(\theta)\atop 2\lambda};1-{\mathrm{e}}^{-2\mathrm{i}\theta}\right).$

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

## §18.35(ii) Orthogonality

First consider type 2.

 18.35.5 $\int_{-1}^{1}P^{(\lambda)}_{n}\left(x;a,b\right)P^{(\lambda)}_{m}\left(x;a,b% \right)w^{(\lambda)}(x;a,b)\,\mathrm{d}x=\frac{\Gamma\left(2\lambda+n\right)}{% n!\,(\lambda+a+n)}\,\delta_{n,m},$ $a\geq b\geq-a$, $\lambda>0$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial, $\,\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Proved: Ismail (2009, Theorem 5.4.2)(proved) Referenced by: (18.35.6), §18.35(ii), Erratum (V1.2.0) for Equation (18.35.5) Permalink: http://dlmf.nist.gov/18.35.E5 Encodings: TeX, pMML, png Modification (effective with 1.2.0): This equation which was previously only valid for $n\neq m$, was updated to give the full normalization and the constraints which were originally given in (18.35.6) were moved to this equation. The constraint $\lambda>-\frac{1}{2}$ has been updated to be $\lambda>0$. See also: Annotations for §18.35(ii), §18.35 and Ch.18

where

 18.35.6 $w^{(\lambda)}(\cos\theta;a,b)={\pi}^{-1}\*{\mathrm{e}}^{(2\theta-\pi)\*\tau_{a% ,b}(\theta)}\*\left(2\sin\theta\right)^{2\lambda-1}\*{\left|\Gamma\left(% \lambda+\mathrm{i}\tau_{a,b}(\theta)\right)\right|}^{2},$ $0<\theta<\pi$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin\NVar{z}$: sine function, $w(x)$: weight function and $\tau_{a,b}(\theta)$ Source: Ismail (2009, (5.4.13)) Referenced by: (18.35.5), §18.35(ii), §18.39(iv), §18.39(iv), Erratum (V1.2.0) for Equation (18.35.5) Permalink: http://dlmf.nist.gov/18.35.E6 Encodings: TeX, pMML, png Modification (effective with 1.2.0): The constraints in this equation have been moved to (18.35.5) except for $0<\theta<\pi$. See also: Annotations for §18.35(ii), §18.35 and Ch.18

Note that

 18.35.6_1 $\ln\left(w^{(\lambda)}(\cos\theta;a,b)\right)=\begin{cases}-2\pi(a+b)\theta^{-% 1}+(2\lambda-1)\ln\left(a+b\right)+\lambda\ln 4+2(a+b)+O\left(\theta\right),&% \theta\to 0+,\\ 2\pi(b-a)\left(\pi-\theta\right)^{-1}+(2\lambda-1)\ln\left(a-b\right)+\lambda% \ln 4+2(a-b)+O\left(\pi-\theta\right),&\theta\to\pi-,\end{cases}$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\ln\NVar{z}$: principal branch of logarithm function and $w(x)$: weight function Referenced by: §18.35(ii), Erratum (V1.2.0) §18.35 Permalink: http://dlmf.nist.gov/18.35.E6_1 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(ii), §18.35 and Ch.18

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

More generally, the $P^{(\lambda)}_{n}\left(x;a,b\right)$ 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 $\mathrm{(i)}\;\lambda>0\mbox{ and }a+\lambda>0,\quad\mathrm{(ii)}\;-\tfrac{1}{% 2}<\lambda<0\mbox{ and }-1 ⓘ Symbols: $n$: nonnegative integer and $x$: real variable Proved: Ismail (2009, (5.5.8)); Case (iii) is overlooked in the given reference.(proved) Proof sketch: By Favard’s theorem (see §18.2(viii)) the necessary and sufficient condition for orthogonality is that in (18.35.2_4), for $c=0$, the coefficient of $Q^{(\lambda)}_{n}(x;a,b,0)$ is real for $n\geq 0$ and the coefficient of $Q^{(\lambda)}_{n-1}(x;a,b,0)$ is positive for $n\geq 1$. Referenced by: §18.35(ii), Erratum (V1.2.0) for Equation (18.35.5) Permalink: http://dlmf.nist.gov/18.35.E6_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(ii), §18.35 and Ch.18

Then

 18.35.6_3 ${\int_{-1}^{1}P^{(\lambda)}_{n}\left(x;a,b\right)P^{(\lambda)}_{m}\left(x;a,b% \right)w^{(\lambda)}(x;a,b)\,\mathrm{d}x+\sum_{\zeta\in D}P^{(\lambda)}_{n}% \left(\zeta;a,b\right)P^{(\lambda)}_{m}\left(\zeta;a,b\right)w_{\zeta}^{(% \lambda)}(a,b)=\frac{\Gamma\left(2\lambda+n\right)}{n!\,(\lambda+a+n)}\,\delta% _{n,m},}$

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 $b>a\geq-b$, $\lambda+a>0$ then

 18.35.6_4 $\displaystyle D$ $\displaystyle=\{x_{k}=\frac{(\lambda+k)\Delta-ab}{a^{2}-\left(\lambda+k\right)% ^{2}}\}_{k=0}^{\infty},$ $\displaystyle w_{x_{k}}^{(\lambda)}(a,b)$ $\displaystyle=\frac{\rho^{2k-1}\left(1-\rho^{2}\right)^{2\lambda+1}\Gamma\left% (2\lambda+k\right)}{2\Delta k!},$ $\displaystyle\Delta$ $\displaystyle=\sqrt{\left(\lambda+k\right)^{2}+b^{2}-a^{2}},$ $\displaystyle\rho$ $\displaystyle=\frac{\Delta-b}{\lambda+k-a},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $!$: factorial (as in $n!$), $(\NVar{a},\NVar{b})$: open interval, $w_{x}$: weights, $k$: nonnegative integer and $x$: real variable Notes: The original source Ismail (2009, §5.5) contains a typo and the notation has been simplified. Note that $\rho=-\rho_{1}(x_{k})$ in the original source. Referenced by: §18.39(iv), Erratum (V1.2.0) for Equation (18.35.5) Permalink: http://dlmf.nist.gov/18.35.E6_4 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(ii), §18.35 and Ch.18

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

For type 3 orthogonality (18.35.5) generalizes to

 18.35.6_5 ${\int_{-1}^{1}P^{{(\lambda)}}_{n}\left(x;a,b,c\right)P^{{(\lambda)}}_{m}\left(% x;a,b,c\right)w^{(\lambda)}(x;a,b,c)\,\mathrm{d}x=\frac{\Gamma\left(c+1\right)% \Gamma\left(2\lambda+c+n\right)}{{\left(c+1\right)_{n}}(\lambda+a+c+n)}\,% \delta_{n,m},}$

where

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

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

## §18.35(iii) Other Properties

 18.35.7 $(1-z{\mathrm{e}}^{\mathrm{i}\theta})^{-\lambda+\mathrm{i}\tau_{a,b}(\theta)}(1% -z{\mathrm{e}}^{-\mathrm{i}\theta})^{-\lambda-\mathrm{i}\tau_{a,b}(\theta)}=% \sum_{n=0}^{\infty}P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)z^{n},$ $|z|<1$, $0<\theta<\pi$.
 18.35.8 $P^{(\lambda)}_{n}\left(x;0,0\right)=C^{(\lambda)}_{n}\left(x\right),$ ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable Proved: Ismail (2009, (5.4.3))(proved) Referenced by: §18.30(viii), §18.35(i) Permalink: http://dlmf.nist.gov/18.35.E8 Encodings: TeX, pMML, png See also: Annotations for §18.35(iii), §18.35 and Ch.18
 18.35.9 $\displaystyle P^{(\lambda)}_{n}\left(x;\phi\right)$ $\displaystyle=P^{(\lambda)}_{n}\left(\cos\phi;0,x\sin\phi\right),$ $\displaystyle P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)$ $\displaystyle=P^{(\lambda)}_{n}\left(\tau_{a,b}(\theta);\theta\right),$ ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$: Meixner–Pollaczek polynomial, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $n$: nonnegative integer, $x$: real variable and $\tau_{a,b}(\theta)$ Proved: Szegő (1975, Appendix, (4.4)); This reference is for the second identity.(proved) Proof sketch: Compare (18.20.10) with (18.35.4_5). Referenced by: §18.35(iii), Erratum (V1.2.0) for Equation (18.35.9) Permalink: http://dlmf.nist.gov/18.35.E9 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.2.0): The second identity was added. See also: Annotations for §18.35(iii), §18.35 and Ch.18
 18.35.10 ${\mathscr{P}}^{\lambda}_{n}\left(x;\phi,c\right)=P^{{(\lambda)}}_{n}\left(\cos% \phi;0,x\sin\phi,c\right).$ ⓘ Symbols: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$: Pollaczek polynomial, ${\mathscr{P}}^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{\phi},\NVar{c}\right)$: associated Meixner–Pollaczek polynomial, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $n$: nonnegative integer and $x$: real variable Source: Askey and Wimp (1984) Referenced by: §18.30(v), §18.35(iii), Erratum (V1.2.0) §18.35 Permalink: http://dlmf.nist.gov/18.35.E10 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.35(iii), §18.35 and Ch.18

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

See Bo and Wong (1996) for an asymptotic expansion of $P^{(\frac{1}{2})}_{n}\left(\cos\left(n^{-\frac{1}{2}}\theta\right);a,b\right)$ as $n\to\infty$, with $a$ and $b$ fixed. This expansion is in terms of the Airy function $\operatorname{Ai}\left(x\right)$ 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 $P^{(\frac{1}{2})}_{n}\left(\cos\left(n^{-\frac{1}{2}}\theta\right);a,b\right)$.

See Szegő (1975, Appendix, §§ 1–5), Askey (1982b), and Ismail (2009, §§ 5.4–5.5) for further results on type 2 Pollaczek polynomials. These polynomials also occur in connection with the Coulomb problem, see §18.39(iv).