Pollaczek polynomials

SearchAdvancedHelp

(0.003 seconds)

1—10 of 20 matching pages

1: 18.35 Pollaczek Polynomials

§18.35 Pollaczek Polynomials

… ►

$P_{-1}^{(\lambda )}(x;a,b,c)=0,$

►

$P_0^{(\lambda )}(x;a,b,c)=1,$

… ►

$Q_{-1}^{(\lambda )}(x;a,b,c)=0,$

►

$Q_0^{(\lambda )}(x;a,b,c)=1,$

…

2: 18.24 Hahn Class: Asymptotic Approximations

… ►For an asymptotic expansion of $P_n^{(\lambda )}(nx;\varphi )$ as $n\to \mathrm{\infty }$, with $\varphi$ fixed, see Li and Wong (2001). …Corresponding approximations are included for the zeros of $P_n^{(\lambda )}(nx;\varphi )$. … ►For asymptotic approximations to $P_n^{(\lambda )}(x;\varphi )$ as $|x+i\lambda |\to \mathrm{\infty }$, with $n$ fixed, see Temme and López (2001). …

3: 18.21 Hahn Class: Interrelations

… ►

18.21.10 $$\underset{t\to \mathrm{\infty }}{lim}{t}^{-n}{p}_n(x-t;\lambda +it,-t\tan \varphi ,\lambda -it,-t\tan \varphi )=\frac{{(-1)}^n}{{(\cos \varphi )}^n}{P}_n^{(\lambda )}(x;\varphi ).$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $\cos z$: cosine function, $\tan z$: tangent function, $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.22(ii), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.21.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

►

18.21.11 $$p_n(x;a,a+\frac{1}{2},a,a+\frac{1}{2})=2^{-2n}{\left(4a+n\right)}_n{P}_n^{(2a)}(2x;\frac{1}{2}\pi ).$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $p_n(x;a,b,\overline{a},\overline{b})$: continuous Hahn polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii)

Permalink:

http://dlmf.nist.gov/18.21.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

►

Meixner–Pollaczek $\to$ Laguerre

►

18.21.12 $$\underset{\varphi \to 0}{lim}{P}_n^{(\frac{1}{2}\alpha +\frac{1}{2})}(-{(2\varphi )}^{-1}x;\varphi )=L_n^{(\alpha )}\left(x\right).$$

ⓘ

Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.24, §18.30(v)

Permalink:

http://dlmf.nist.gov/18.21.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

… ►

18.21.13 $$n!\underset{\lambda \to \mathrm{\infty }}{lim}{\lambda }^{-n/2}{P}_n^{(\lambda )}(x{\lambda }^{1/2};\pi /2)=H_n\left(x\right).$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.21(ii), §18.30(v), Erratum (V1.2.0) Section 18.21

Permalink:

http://dlmf.nist.gov/18.21.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

…

4: 18.22 Hahn Class: Recurrence Relations and Differences

… ►

18.22.7 $$p_n(x)=P_n^{(\lambda )}(x;\varphi ),$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(i)

Permalink:

http://dlmf.nist.gov/18.22.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

… ►

18.22.16 $$p_n(x)=P_n^{(\lambda )}(x;\varphi ),$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(ii)

Permalink:

http://dlmf.nist.gov/18.22.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

… ►

18.22.29 $${\delta }_x\left(P_n^{(\lambda )}(x;\varphi )\right)=2\sin \varphi P_{n-1}^{(\lambda +\frac{1}{2})}(x;\varphi ),$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\delta }_x$: central difference, $\sin z$: sine function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

►

18.22.30 $${\delta }_x\left(w^{(\lambda +\frac{1}{2})}(x;\varphi )P_n^{(\lambda +\frac{1}{2})}(x;\varphi )\right)=-(n+1)w^{(\lambda )}(x;\varphi )P_{n+1}^{(\lambda )}(x;\varphi ).$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\delta }_x$: central difference, $w(x)$: weight function, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i), §18.22(iii)

Permalink:

http://dlmf.nist.gov/18.22.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

5: 18.19 Hahn Class: Definitions

… ► ►

18.19.6 $$p_n(x)=P_n^{(\lambda )}(x;\varphi ),$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19

Permalink:

http://dlmf.nist.gov/18.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §18.19, §18.19 and Ch.18

…

6: 18.20 Hahn Class: Explicit Representations

… ►

18.20.4 $$w^{(\lambda )}(x;\varphi )P_n^{(\lambda )}(x;\varphi )=\frac{1}{n!}{\delta }_x^n\left(w^{(\lambda +\frac{1}{2}n)}(x;\varphi )\right).$$

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\delta }_x$: central difference, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.20(i)

Permalink:

http://dlmf.nist.gov/18.20.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(i), §18.20(i), §18.20 and Ch.18

… ►

18.20.10 $$P_n^{(\lambda )}(x;\varphi )=\frac{{\left(2\lambda \right)}_n}{n!}{\mathrm{e}}^{\mathrm{i}n\varphi }{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,\lambda +\mathrm{i}x}{2\lambda };1-{\mathrm{e}}^{-2\mathrm{i}\varphi }).$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.19, §18.20(ii), §18.21(ii), §18.22(iii), §18.26(ii), (18.35.9)

Permalink:

http://dlmf.nist.gov/18.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.20(ii), §18.20 and Ch.18

7: 18.1 Notation

… ►

$${(z_1,\mathrm{\dots },z_k;q)}_{\mathrm{\infty }}={(z_1;q)}_{\mathrm{\infty }}\mathrm{\cdots }{(z_k;q)}_{\mathrm{\infty }}.$$

… ►

Meixner–Pollaczek: $P_n^{(\lambda )}(x;\varphi )$.

… ►

Pollaczek: $P_n^{(\lambda )}(x;a,b)$, $P_n^{(\lambda )}(x;a,b,c)$.

…

8: 18.23 Hahn Class: Generating Functions

… ►

18.23.7 $${(1-{\mathrm{e}}^{\mathrm{i}\varphi }z)}^{-\lambda +\mathrm{i}x}{(1-{\mathrm{e}}^{-\mathrm{i}\varphi }z)}^{-\lambda -\mathrm{i}x}=\sum _{n=0}^{\mathrm{\infty }}{P}_n^{(\lambda )}(x;\varphi )z^n,$$ .

ⓘ

Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii), §18.23

Permalink:

http://dlmf.nist.gov/18.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

9: 18.30 Associated OP’s

… ►

§18.30(v) Associated Meixner–Pollaczek Polynomials

►In view of (18.22.8) the associated Meixner–Pollaczek polynomials ${𝒫}_n^{\lambda }(x;\varphi ,c)$ are defined by the recurrence relation ►

${𝒫}_{-1}^{\lambda }(x;\varphi ,c)=0,$

… ►The type 3 Pollaczek polynomials are the associated type 2 Pollaczek polynomials, see §18.35. The relationship (18.35.8) implies the definition for the associated ultraspherical polynomials $C_n^{(\lambda )}(x;c)=P_n^{(\lambda )}(x;0,0,c)$. …

10: 18.39 Applications in the Physical Sciences

… ►Thus the $c_N(x)=P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$ and the eigenvalues …are determined by the $N$ zeros, $x_i^N$ of the Pollaczek polynomial $P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$. ►

The Coulomb–Pollaczek Polynomials

►The polynomials $P_N^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s})$, for both positive and negative $Z$, define the Coulomb–Pollaczek polynomials (CP OP’s in what follows), see Yamani and Reinhardt (1975, Appendix B, and §IV). … ►

18.39.54 $${\mathrm{\Psi }}_{x,l}(r)=B_l(x)\sum _{n=0}^{\mathrm{\infty }}\frac{n!}{\mathrm{\Gamma }(n+2l+2)}{P}_n^{(l+1)}(x;\frac{2Z}{s},-\frac{2Z}{s}){\varphi }_{n,l}(sr),$$ $x=\frac{8ϵ-s^2}{8ϵ+s^2}$,

ⓘ

Symbols:

$P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $!$: factorial (as in $n!$), $l$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.39.E54

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(iv), §18.39(iv), §18.39 and Ch.18

…