18 Orthogonal PolynomialsAskey Scheme18.20 Hahn Class: Explicit Representations18.22 Hahn Class: Recurrence Relations and Differences

§18.21 Hahn Class: Interrelations

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Keywords:

Hahn class orthogonal polynomials, interrelations with other orthogonal polynomials

Referenced by:

Erratum (V1.2.0) §18.21

Permalink:

http://dlmf.nist.gov/18.21

See also:

Annotations for Ch.18

Contents

1. §18.21(i) Dualities
2. §18.21(ii) Limit Relations and Special Cases

§18.21(i) Dualities

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Keywords:

Hahn class orthogonal polynomials, Wilson class orthogonal polynomials, dualities

Permalink:

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

See also:

Annotations for §18.21 and Ch.18

Duality of Hahn and Dual Hahn

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See also:

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

18.21.1		$$Q_n(x;\alpha ,\beta ,N)=R_x(n(n+\alpha +\beta +1);\alpha ,\beta ,N),$$
$n,x=0,1,\mathrm{\dots },N$.
ⓘ Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii), §18.26(i), §18.38(iii) Permalink: http://dlmf.nist.gov/18.21.E1 Encodings: TeX, pMML, png See also: Annotations for §18.21(i), §18.21(i), §18.21 and Ch.18

For the dual Hahn polynomial $R_n(x;\gamma ,\delta ,N)$ see §18.25.

Self-Dualities

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See also:

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

18.21.2		$K_n(x;p,N)$	$=K_x(n;p,N),$
	$n,x=0,1,\mathrm{\dots },N$.
		$M_n(x;\beta ,c)$	$=M_x(n;\beta ,c),$
	$n,x=0,1,2,\mathrm{\dots }$.
		$C_n(x;a)$	$=C_x(n;a),$
	$n,x=0,1,2,\mathrm{\dots }$.
ⓘ Symbols: $C_n(x;a)$: Charlier polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii), §18.22(ii) Permalink: http://dlmf.nist.gov/18.21.E2 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §18.21(i), §18.21(i), §18.21 and Ch.18

§18.21(ii) Limit Relations and Special Cases

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Keywords:

Hahn class orthogonal polynomials, limit relations, special cases

Notes:

For (18.21.3), (18.21.5), (18.21.7), (18.21.8) see Ismail (2009, §6.2, unnumbered formula after (6.2.34), also (6.2.17), (6.1.19), (6.1.18)). For (18.21.1) see Karlin and McGregor (1961, (1.19)). The three identities in (18.21.2) follow from (18.20.6), (18.20.7), (18.20.8). (18.21.4) follows from (18.20.5) and (18.20.7). (18.21.6) follows from (18.20.6) and (18.20.8). (18.21.9) follows from (18.22.2), Row 4 in Table 18.22.1, (18.9.1), and Row 10 in Table 18.9.1. (18.21.10) follows from (18.20.9) and (18.20.10). For (18.21.11) see Koornwinder (1989, (2.6)). (18.21.12) follows from (18.20.10) and (18.5.12). For Figure 18.21.1 see Askey and Wilson (1985, p. 46), together with correction in Askey (1985).

Referenced by:

§18.19, §18.21(ii)

Permalink:

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

Addition (effective with 1.2.0):

Equation (18.21.13) was added.

See also:

Annotations for §18.21 and Ch.18

Hahn $\to$ Krawtchouk

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See also:

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

18.21.3		$$\underset{t\to \mathrm{\infty }}{lim}{Q}_n(x;pt,(1-p)t,N)=K_n(x;p,N).$$
ⓘ Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $t$: real variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii) Permalink: http://dlmf.nist.gov/18.21.E3 Encodings: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Hahn $\to$ Meixner

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See also:

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

18.21.4		$$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(x;\beta -1,N(c^{-1}-1),N)=M_n(x;\beta ,c).$$
ⓘ Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii) Permalink: http://dlmf.nist.gov/18.21.E4 Encodings: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Hahn $\to$ Jacobi

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Keywords:

Jacobi polynomials, interrelations with other orthogonal polynomials

See also:

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

18.21.5		$$\underset{N\to \mathrm{\infty }}{lim}{Q}_n(Nx;\alpha ,\beta ,N)=\frac{P_n^{(\alpha ,\beta )}\left(1-2x\right)}{P_n^{(\alpha ,\beta )}\left(1\right)}.$$
ⓘ Symbols: $Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii) Permalink: http://dlmf.nist.gov/18.21.E5 Encodings: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Krawtchouk $\to$ Charlier

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See also:

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

18.21.6		$$\underset{N\to \mathrm{\infty }}{lim}{K}_n(x;N^{-1}a,N)=C_n(x;a).$$
ⓘ Symbols: $C_n(x;a)$: Charlier polynomial, $K_n(x;p,N)$: Krawtchouk polynomial, $N$: positive integer, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii) Permalink: http://dlmf.nist.gov/18.21.E6 Encodings: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Meixner $\to$ Charlier

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See also:

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

18.21.7		$$\underset{\beta \to \mathrm{\infty }}{lim}{M}_n(x;\beta ,a{(a+\beta )}^{-1})=C_n(x;a).$$
ⓘ Symbols: $C_n(x;a)$: Charlier polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii), §18.22(iii) Permalink: http://dlmf.nist.gov/18.21.E7 Encodings: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Meixner $\to$ Laguerre

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Keywords:

Laguerre polynomials, interrelations with other orthogonal polynomials

See also:

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

18.21.8		$$\underset{c\to 1}{lim}{M}_n({(1-c)}^{-1}x;\alpha +1,c)=\frac{L_n^{(\alpha )}\left(x\right)}{L_n^{(\alpha )}\left(0\right)}.$$
ⓘ Symbols: $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $M_n(x;\beta ,c)$: Meixner polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii) Permalink: http://dlmf.nist.gov/18.21.E8 Encodings: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Charlier $\to$ Hermite

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Keywords:

Hermite polynomials, interrelations with other orthogonal polynomials

See also:

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

18.21.9		$$\underset{a\to \mathrm{\infty }}{lim}{(2a)}^{\frac{1}{2}n}{C}_n({(2a)}^{\frac{1}{2}}x+a;a)={(-1)}^n{H}_n\left(x\right).$$
ⓘ Symbols: $C_n(x;a)$: Charlier polynomial, $H_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.21(ii) Permalink: http://dlmf.nist.gov/18.21.E9 Encodings: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Continuous Hahn $\to$ Meixner–Pollaczek

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See also:

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

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: TeX, pMML, png 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: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Meixner–Pollaczek $\to$ Laguerre

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Keywords:

Laguerre polynomials, interrelations with other orthogonal polynomials

See also:

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

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: TeX, pMML, png See also: Annotations for §18.21(ii), §18.21(ii), §18.21 and Ch.18

Meixner–Pollaczek $\to$ Hermite

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Addition (effective with 1.2.0):

This paragraph as well (18.21.13) were added.

See also:

Annotations for §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) §18.21 Permalink: http://dlmf.nist.gov/18.21.E13 Encodings: TeX, pMML, png 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

A graphical representation of limits in §§18.7(iii), 18.21(ii), and 18.26(ii) is provided by the Askey scheme depicted in Figure 18.21.1.

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Keywords:

Askey scheme for orthogonal polynomials, Askey–Wilson class orthogonal polynomials, Hahn class orthogonal polynomials, Hermite polynomials, Jacobi polynomials, Laguerre polynomials, Wilson class orthogonal polynomials, classical orthogonal polynomials, interrelations, interrelations with other orthogonal polynomials, with other orthogonal polynomials

Referenced by:

§18.19, §18.21(ii), §18.21(ii), §18.26(ii), §18.7(iii)

Permalink:

http://dlmf.nist.gov/18.21.F1

Encodings:

pdf, png

See also:

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

18.20 Hahn Class: Explicit Representations18.22 Hahn Class: Recurrence Relations and Differences

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