18 Orthogonal Polynomials Other Orthogonal Polynomials 18.34 Bessel Polynomials 18.36 Miscellaneous Polynomials

§18.35 Pollaczek Polynomials

ⓘ

Keywords:: Pollaczek polynomials
Referenced by:: §18.30(viii), Erratum (V1.2.0) §18.35
Permalink:: http://dlmf.nist.gov/18.35
See also:: Annotations for Ch.18

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

§18.35(i) Definition and Hypergeometric Representation

ⓘ

Keywords:: Pollaczek polynomials, as associated type 2 Pollaczek polynomials, hypergeometric function, of type 1, of type 2, of type 3, recurrence relation, relation to hypergeometric function, relations to other functions, type 2 Pollaczek polynomials
Permalink:: http://dlmf.nist.gov/18.35.i
Addition (effective with 1.2.0):: Equations (18.35.0_5), (18.35.2_1)–(18.35.2_5), and (18.35.4_5) were added. A remark about type 1 Pollaczek polynomials was added underneath (18.35.4_5).
Update (effective with 1.2.0):: This recurrence relation which was previously given for Pollaczek polynomials of type 2 has been updated for Pollaczek polynomials of type 3.
See also:: Annotations for §18.35 and Ch.18

There are 3 types of Pollaczek polynomials:

18.35.0_5

$P^{(\frac{1}{2})}_{n}\left(x;a,b\right),$

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

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

ⓘ

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, : nonnegative integer and : 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 reduces to type 2, and type 3 with 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 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

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

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

ⓘ

Symbols:: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : Pollaczek polynomial and : 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 $P^{{(\lambda)}}_{n+1}\left(x;a,b,c\right)=\frac{2(n+c+\lambda+a)x+2b}{n+c+1}\,% P^{{(\lambda)}}_{n}\left(x;a,b,c\right)-\frac{n+c+2\lambda-1}{n+c+1}\,P^{{(% \lambda)}}_{n-1}\left(x;a,b,c\right),$ $n=0,1,\dots$ ,

ⓘ

Symbols:: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : Pollaczek polynomial, : nonnegative integer and : 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, : nonnegative integer and : 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), : nonnegative integer and : 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

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

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

ⓘ

Symbols:: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : Pollaczek polynomial and : 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 $x{Q}^{{(\lambda)}}_{n}\left(x;a,b,c\right)={Q}^{{(\lambda)}}_{n+1}\left(x;a,b,% c\right)-\frac{b}{n+c+\lambda+a}{Q}^{{(\lambda)}}_{n}\left(x;a,b,c\right)+% \frac{(n+c)(n+c+2\lambda-1)}{4(n+c+\lambda+a-1)(n+c+\lambda+a)}{Q}^{{(\lambda)% }}_{n-1}\left(x;a,b,c\right),$ $n=0,1,\dots$ .

ⓘ

Symbols:: ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : Pollaczek polynomial, : nonnegative integer and : 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, : nonnegative integer and : 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 and only occur in the form , 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 ), $\mathrm{i}$ : imaginary unit, $\ell$ : nonnegative integer, : nonnegative integer, : 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 $P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=\frac{{\left(2\lambda\right)_{n}}% }{n!}\,{\mathrm{e}}^{\mathrm{i}n\theta}\*F\left({-n,\lambda+\mathrm{i}\tau_{a,% b}(\theta)\atop 2\lambda};1-{\mathrm{e}}^{-2\mathrm{i}\theta}\right).$

ⓘ

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), $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$ : Pollaczek polynomial, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\mathrm{i}$ : imaginary unit, : nonnegative integer and $\tau_{a,b}(\theta)$
Source:: Erdélyi et al. (1953b, 10.21(10))
Proof sketch:: Apply (15.8.7) to (18.35.4).
Referenced by:: (18.35.9), §18.35(i), Erratum (V1.2.0) §18.35
Permalink:: http://dlmf.nist.gov/18.35.E4_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

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

§18.35(ii) Orthogonality

ⓘ

Keywords:: Pollaczek polynomials, Szegő class, not in Szegő class, of type 2, of type 3, orthogonality properties, type 2 Pollaczek polynomials counterexample
Permalink:: http://dlmf.nist.gov/18.35.ii
Addition (effective with 1.2.0):: Equations (18.35.6_1)–(18.35.6_6) were added as well as accompanying extensive text below (18.35.6).
See also:: Annotations for §18.35 and Ch.18

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 , : factorial (as in ), $\int$ : integral, : weight function, : nonnegative integer, : nonnegative integer and : 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, : 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 : 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 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 $\mathrm{(i)}\;\lambda>0\mbox{ and }a+\lambda>0,\quad\mathrm{(ii)}\;-\tfrac{1}{% 2}<\lambda<0\mbox{ and }-1<a+\lambda<0,\quad\mathrm{(iii)}\;\lambda=0\mbox{ % and }a=b=0.$

ⓘ

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

ⓘ

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 , $\in$ : element of, : factorial (as in ), $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, : weight function, $w_{x}$ : weights, : nonnegative integer, : nonnegative integer and : real variable
Proved:: Ismail (2009, (5.5.18))(proved)
Referenced by:: §18.39(iv)
Permalink:: http://dlmf.nist.gov/18.35.E6_3
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

where, depending on $a,b,\lambda$ , 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 and condition (ii) of (18.35.6_2) holds then (see Ismail (2009, Theorem 5.5.1)). Also, if $b>a\geq-b$ , $\lambda+a>0$ then

18.35.6_4

$D=\Bigg\{x_{k}=\frac{(\lambda+k)\Delta-ab}{a^{2}-\left(\lambda+k\right)^{2}}% \Bigg\}_{k=0}^{\infty},$

$w_{x_{k}}^{(\lambda)}(a,b)=\frac{\rho^{2k-1}\left(1-\rho^{2}\right)^{2\lambda+% 1}\Gamma\left(2\lambda+k\right)}{2\Delta k!},$

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

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

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, : factorial (as in ), $(\NVar{a},\NVar{b})$ : open interval, $w_{x}$ : weights, : nonnegative integer and : 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 ${\int_{-1}^{1}P^{{(\lambda)}}_{n}\left(x;a,b,c\right)P^{{(\lambda)}}_{m}\left(% x;a,b,c\right)w^{(\lambda)}(x;a,b,c)\,\mathrm{d}x=\frac{\Gamma\left(c+1\right)% \Gamma\left(2\lambda+c+n\right)}{{\left(c+1\right)_{n}}(\lambda+a+c+n)}\,% \delta_{n,m},}$

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), ${Q}^{{(\NVar{\lambda})}}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c}\right)$ : Pollaczek polynomial, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : weight function, : nonnegative integer, : nonnegative integer and : real variable
Proved:: Pollaczek (1950, (3))(proved)
Permalink:: http://dlmf.nist.gov/18.35.E6_5
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

where

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

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ 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, $\pi$ : 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, : weight function and $\tau_{a,b}(\theta)$
Source:: Pollaczek (1950, (2))
Referenced by:: §18.35(ii), Erratum (V1.2.0) §18.35
Permalink:: http://dlmf.nist.gov/18.35.E6_6
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

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

§18.35(iii) Other Properties

ⓘ

Keywords:: Meixner–Pollaczek polynomials, Pollaczek polynomials, generating function, of type 2, relation to type 2 Pollaczek polynomials, relations to other functions, relations to other orthogonal polynomials, type 2 Pollaczek polynomials, ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.35.iii
Addition (effective with 1.2.0):: One extra identity was added to (18.35.9), and equation (18.35.10) is new. One extra sentence was added to the end of the subsection.
See also:: Annotations for §18.35 and Ch.18

18.35.7 $(1-z{\mathrm{e}}^{\mathrm{i}\theta})^{-\lambda+\mathrm{i}\tau_{a,b}(\theta)}(1% -z{\mathrm{e}}^{-\mathrm{i}\theta})^{-\lambda-\mathrm{i}\tau_{a,b}(\theta)}=% \sum_{n=0}^{\infty}P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)z^{n},$

, $0<\theta<\pi$ .

ⓘ

Symbols:: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$ : Pollaczek polynomial, $\pi$ : 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, : complex variable, : nonnegative integer and $\tau_{a,b}(\theta)$
Proved:: Ismail (2009, (5.4.8))(proved)
Permalink:: http://dlmf.nist.gov/18.35.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §18.35(iii), §18.35 and Ch.18

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, : nonnegative integer and : 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

$P^{(\lambda)}_{n}\left(x;\phi\right)=P^{(\lambda)}_{n}\left(\cos\phi;0,x\sin% \phi\right),$

$P^{(\lambda)}_{n}\left(\cos\theta;a,b\right)=P^{(\lambda)}_{n}\left(\tau_{a,b}% (\theta);\theta\right),$

ⓘ

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, : nonnegative integer, : 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, : nonnegative integer and : 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 and 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).

18.34 Bessel Polynomials 18.36 Miscellaneous Polynomials

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§18.35 Pollaczek Polynomials

Contents

§18.35(i) Definition and Hypergeometric Representation

§18.35(ii) Orthogonality

§18.35(iii) Other Properties