Meixner polynomials

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1: 18.21 Hahn Class: Interrelations

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$M_n(x;\beta ,c)=M_x(n;\beta ,c),$ $n,x=0,1,2,\mathrm{\dots }$.

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

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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

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

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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

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

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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

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

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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

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2: 18.24 Hahn Class: Asymptotic Approximations

… ►For two asymptotic expansions of $M_n(nx;\beta ,c)$ as $n\to \mathrm{\infty }$, with $\beta$ and $c$ fixed, see Jin and Wong (1998) and Wang and Wong (2011). … ►For asymptotic approximations for the zeros of $M_n(nx;\beta ,c)$ in terms of zeros of $\mathrm{Ai}\left(x\right)$ (§9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … ►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: 18.22 Hahn Class: Recurrence Relations and Differences

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

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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

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Table 18.22.2: Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials.

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$p_n(x)$	$A(x)$	$C(x)$	${\lambda }_n$
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$M_n(x;\beta ,c)$	$c(x+\beta )$	$x$	$n(1-c)$
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18.22.16 $$p_n(x)=P_n^{(\lambda )}(x;\varphi ),$$

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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

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

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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

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

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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

4: 18.19 Hahn Class: Definitions

… ►Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_n(x;\alpha ,\beta ,N)$, Krawtchouk polynomials $K_n(x;p,N)$, Meixner polynomials $M_n(x;\beta ,c)$, and Charlier polynomials $C_n(x;a)$. ►

Table 18.19.1: Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s: discrete sets, weight functions, standardizations, and parameter constraints.

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	$p_n(x)$	$X$	$w_x$	$h_n$
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Meixner	$M_n(x;\beta ,c)$	$\{0,1,2,\mathrm{\dots }\}$	${\left(\beta \right)}_x{c}^x/x!$, $\beta >0$,	${\displaystyle \frac{c^{-n}n!}{{\left(\beta \right)}_n{(1-c)}^{\beta }}}$
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Table 18.19.2: Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients.

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$p_n(x)$	$k_n$
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$M_n(x;\beta ,c)$	${(1-c^{-1})}^n/{\left(\beta \right)}_n$
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18.19.6 $$p_n(x)=P_n^{(\lambda )}(x;\varphi ),$$

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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

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5: 18.20 Hahn Class: Explicit Representations

… ►For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ and ${\kappa }_n$ are as in Table 18.20.1. ►

Table 18.20.1: Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1).

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$p_n(x)$	$F(x)$	${\kappa }_n$
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$M_n(x;\beta ,c)$	$x+\beta$	${\left(\beta \right)}_n$
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18.20.4 $$w^{(\lambda )}(x;\varphi )P_n^{(\lambda )}(x;\varphi )=\frac{1}{n!}{\delta }_x^n\left(w^{(\lambda +\frac{1}{2}n)}(x;\varphi )\right).$$

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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

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18.20.7 $$M_n(x;\beta ,c)={}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-n,-x}{\beta };1-c^{-1}).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $M_n(x;\beta ,c)$: Meixner polynomial, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.21(ii), §18.26(ii)

Permalink:

http://dlmf.nist.gov/18.20.E7

Encodings:

pMML, png, TeX

See also:

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

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

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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

6: 18.23 Hahn Class: Generating Functions

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18.23.4 $${\left(1-\frac{z}{c}\right)}^x{(1-z)}^{-x-\beta }=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\beta \right)}_n}{n!}{M}_n(x;\beta ,c)z^n,$$ $x=0,1,2,\mathrm{\dots }$, .

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Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.2(xii)

Permalink:

http://dlmf.nist.gov/18.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.23, §18.23 and Ch.18

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

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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

7: 18.30 Associated OP’s

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§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 ►

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

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${𝒫}_0^{\lambda }(x;\varphi ,c)=1,$

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$(n+c+1){𝒫}_{n+1}^{\lambda }(x;\varphi ,c)=(2x\sin \varphi +2(n+c+\lambda )\cos \varphi ){𝒫}_n^{\lambda }(x;\varphi ,c)-(n+c+2\lambda -1){𝒫}_{n-1}^{\lambda }(x;\varphi ,c),$ $n=0,1,\mathrm{\dots }$.

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8: 18.35 Pollaczek Polynomials

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$P_n^{(\lambda )}(x;\varphi )=P_n^{(\lambda )}(\cos \varphi ;0,x\sin \varphi ),$

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$P_n^{(\lambda )}(\cos \theta ;a,b)=P_n^{(\lambda )}({\tau }_{a,b}(\theta );\theta ),$

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18.35.10 $${𝒫}_n^{\lambda }(x;\varphi ,c)=P_n^{(\lambda )}(\cos \varphi ;0,x\sin \varphi ,c).$$

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Symbols:

$Q_n^{(\lambda )}(x;a,b,c)$: Pollaczek polynomial, ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\cos z$: cosine function, $\sin z$: 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) Section 18.35

Permalink:

http://dlmf.nist.gov/18.35.E10

Encodings:

pMML, png, TeX

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 $C_n^{(\lambda )}\left(x\right)$, the Meixner–Pollaczek polynomials $P_n^{(\lambda )}(x;\varphi )$ and the associated Meixner–Pollaczek polynomials ${𝒫}_n^{\lambda }(x;\varphi ,c)$ see §§18.3, 18.19 and 18.30(v), respectively. …

9: 18.1 Notation

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

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Meixner: $M_n(x;\beta ,c)$.

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

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10: 18.26 Wilson Class: Continued

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18.26.8 $$\underset{t\to \mathrm{\infty }}{lim}{S}_n({(x-t)}^2;\lambda +it,\lambda -it,t\cot \varphi )/t^n=n!{(\csc \varphi )}^n{P}_n^{(\lambda )}(x;\varphi ).$$

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Symbols:

$P_n^{(\lambda )}(x;\varphi )$: Meixner–Pollaczek polynomial, $S_n(x;a,b,c)$: continuous dual Hahn polynomial, $\csc z$: cosecant function, $\cot z$: cotangent function, $!$: factorial (as in $n!$), $t$: real variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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18.26.13 $$\underset{N\to \mathrm{\infty }}{lim}{R}_n(r(x;\beta ,c,N);\beta -1,c^{-1}(1-c)N,N)=M_n(x;\beta ,c).$$

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Symbols:

$M_n(x;\beta ,c)$: Meixner polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $n$: nonnegative integer, $x$: real variable and $r(x;\beta ,c,N)$

Referenced by:

§18.26(ii)

Permalink:

http://dlmf.nist.gov/18.26.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(ii), §18.26(ii), §18.26 and Ch.18

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