18 Orthogonal Polynomials Askey Scheme 18.20 Hahn Class: Explicit Representations 18.22 Hahn Class: Recurrence Relations and Differences

§18.21 Hahn Class: Interrelations

ⓘ

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

§18.21(i) Dualities
§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		$Q_{n}\left(x;\alpha,\beta,N\right)=R_{x}\left(n(n+\alpha+\beta+1);\alpha,\beta% ,N\right),$
$n,x=0,1,\dots,N$ .
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $R_{\NVar{n}}\left(\NVar{x};\NVar{\gamma},\NVar{\delta},\NVar{N}\right)$ : 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 $R_{n}\left(x;\gamma,\delta,N\right)$ see §18.25.

Self-Dualities

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

18.21.2		$\displaystyle K_{n}\left(x;p,N\right)$	$\displaystyle=K_{x}\left(n;p,N\right),$
	$n,x=0,1,\dots,N$ .
		$\displaystyle M_{n}\left(x;\beta,c\right)$	$\displaystyle=M_{x}\left(n;\beta,c\right),$
	$n,x=0,1,2,\dots$ .
		$\displaystyle C_{n}\left(x;a\right)$	$\displaystyle=C_{x}\left(n;a\right),$
	$n,x=0,1,2,\dots$ .
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : 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		$\lim_{t\to\infty}Q_{n}\left(x;pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : 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		$\lim_{N\to\infty}Q_{n}\left(x;\beta-1,N(c^{-1}-1),N\right)=M_{n}\left(x;\beta,% c\right).$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : 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		$\lim_{N\to\infty}Q_{n}\left(Nx;\alpha,\beta,N\right)=\frac{P^{(\alpha,\beta)}_% {n}\left(1-2x\right)}{P^{(\alpha,\beta)}_{n}\left(1\right)}.$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$ : 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		$\lim_{N\to\infty}K_{n}\left(x;N^{-1}a,N\right)=C_{n}\left(x;a\right).$
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : 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		$\lim_{\beta\to\infty}M_{n}\left(x;\beta,a(a+\beta)^{-1}\right)=C_{n}\left(x;a% \right).$
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : 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		$\lim_{c\to 1}M_{n}\left((1-c)^{-1}x;\alpha+1,c\right)=\frac{L^{(\alpha)}_{n}% \left(x\right)}{L^{(\alpha)}_{n}\left(0\right)}.$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : 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		$\lim_{a\to\infty}(2a)^{\frac{1}{2}n}C_{n}\left((2a)^{\frac{1}{2}}x+a;a\right)=% (-1)^{n}H_{n}\left(x\right).$
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $H_{\NVar{n}}\left(\NVar{x}\right)$ : 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		$\lim_{t\to\infty}t^{-n}p_{n}\left(x-t;\lambda+it,-t\tan\phi,\lambda-it,-t\tan% \phi\right)=\frac{(-1)^{n}}{(\cos\phi)^{n}}P^{(\lambda)}_{n}\left(x;\phi\right).$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $\cos\NVar{z}$ : cosine function, $\tan\NVar{z}$ : 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		$p_{n}\left(x;a,a+\tfrac{1}{2},a,a+\tfrac{1}{2}\right)=2^{-2n}{\left(4a+n\right% )_{n}}P^{(2a)}_{n}\left(2x;\tfrac{1}{2}\pi\right).$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\pi$ : the ratio of the circumference of a circle to its diameter, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : 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		$\lim_{\phi\to 0}P^{(\frac{1}{2}\alpha+\frac{1}{2})}_{n}\left(-(2\phi)^{-1}x;% \phi\right)=L^{(\alpha)}_{n}\left(x\right).$
ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$ : Laguerre (or generalized Laguerre) polynomial, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : 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		$n!\lim_{\lambda\to\infty}\lambda^{-n/2}P^{(\lambda)}_{n}\left(x{\lambda}^{1/2}% ;\pi/2\right)=H_{n}\left(x\right).$
ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : 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.

See accompanying text — Figure 18.21.1: Askey scheme. The number of free real parameters is zero for Hermite polynomials. It increases by one for each row ascended in the scheme, culminating with four free real parameters for the Wilson and Racah polynomials. (This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify

18.20 Hahn Class: Explicit Representations 18.22 Hahn Class: Recurrence Relations and Differences

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§18.21 Hahn Class: Interrelations

Contents

§18.21(i) Dualities

Duality of Hahn and Dual Hahn

Self-Dualities

§18.21(ii) Limit Relations and Special Cases

Hahn → Krawtchouk

Hahn → Meixner

Hahn → Jacobi

Krawtchouk → Charlier

Meixner → Charlier

Meixner → Laguerre

Charlier → Hermite

Continuous Hahn → Meixner–Pollaczek

Meixner–Pollaczek → Laguerre

Meixner–Pollaczek → Hermite