18 Orthogonal Polynomials Askey Scheme 18.21 Hahn Class: Interrelations 18.23 Hahn Class: Generating Functions

§18.22 Hahn Class: Recurrence Relations and Differences

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Permalink:: http://dlmf.nist.gov/18.22
See also:: Annotations for Ch.18

§18.22(i) Recurrence Relations in $n$

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Keywords:: Hahn class orthogonal polynomials, recurrence relations
Notes:: For (18.22.1)–(18.22.3) see Ismail (2009, (6.2.8) and (6.2.9)). For Table 18.22.1 see Ismail (2009, (6.2.36), (6.1.5), and (6.1.25)). (18.22.4)–(18.22.6) is a limiting case of Andrews et al. (1999, (3.8.2)), in view of (18.26.6). For (18.22.7)–(18.22.8) see Ismail (2009, (5.9.1)).
Permalink:: http://dlmf.nist.gov/18.22.i
See also:: Annotations for §18.22 and Ch.18

Hahn

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

With

18.22.1		$p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E1 Encodings: TeX, pMML, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

18.22.2		$-xp_{n}(x)=A_{n}p_{n+1}(x)-\left(A_{n}+C_{n}\right)p_{n}(x)+C_{n}p_{n-1}(x),$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable, $A_{n}$ : coefficient and $C_{n}$ : coefficient Referenced by: §16.4(iii), §18.21(ii), §18.22(i), Table 18.22.1 Permalink: http://dlmf.nist.gov/18.22.E2 Encodings: TeX, pMML, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

where

18.22.3	$\displaystyle A_{n}$	$\displaystyle=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2% n+\alpha+\beta+2)},$
18.22.3	$\displaystyle C_{n}$	$\displaystyle=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+% \alpha+\beta+1)}.$
ⓘ Defines: $A_{n}$ : coefficient (locally) and $C_{n}$ : coefficient (locally) Symbols: $N$ : positive integer and $n$ : nonnegative integer Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

Krawtchouk, Meixner, and Charlier

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

These polynomials satisfy (18.22.2) with $p_{n}(x)$ , $A_{n}$ , and $C_{n}$ as in Table 18.22.1.

Table 18.22.1: Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials.

$p_{n}(x)$	$A_{n}$	$C_{n}$
$K_{n}\left(x;p,N\right)$	$p(N-n)$	$n(1-p)$
$M_{n}\left(x;\beta,c\right)$	$\dfrac{c(n+\beta)}{1-c}$	$\dfrac{n}{1-c}$
$C_{n}\left(x;a\right)$	$a$	$n$

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Defines:: $A_{n}$ : coefficient (locally) and $C_{n}$ : coefficient (locally)
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, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.21(ii), §18.22(i), §18.22(i), §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.T1
See also:: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

Continuous Hahn

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

With

18.22.4		$q_{n}(x)=\ifrac{p_{n}\left(x;a,b,\overline{a},\overline{b}\right)}{p_{n}\left(% \mathrm{i}a;a,b,\overline{a},\overline{b}\right)},$
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E4 Encodings: TeX, pMML, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

18.22.5		$(a+\mathrm{i}x)q_{n}(x)=\tilde{A}_{n}q_{n+1}(x)-\bigl{(}\tilde{A}_{n}+\tilde{C% }_{n}\bigr{)}q_{n}(x)+\tilde{C}_{n}q_{n-1}(x),$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $n$ : nonnegative integer, $x$ : real variable, $\tilde{A}_{n}$ : coefficient and $\tilde{C}_{n}$ : coefficient Permalink: http://dlmf.nist.gov/18.22.E5 Encodings: TeX, pMML, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

where

18.22.6	$\displaystyle\tilde{A}_{n}$	$\displaystyle=-\frac{(n+2\Re\left(a+b\right)-1)(n+a+\overline{a})(n+a+% \overline{b})}{(2n+2\Re\left(a+b\right)-1)(2n+2\Re\left(a+b\right))},$
18.22.6	$\displaystyle\tilde{C}_{n}$	$\displaystyle=\frac{n(n+b+\overline{a}-1)(n+b+\overline{b}-1)}{(2n+2\Re\left(a% +b\right)-2)(2n+2\Re\left(a+b\right)-1)}.$
ⓘ Defines: $\tilde{A}_{n}$ : coefficient (locally) and $\tilde{C}_{n}$ : coefficient (locally) Symbols: $\overline{\NVar{z}}$ : complex conjugate, $\Re$ : real part and $n$ : nonnegative integer Referenced by: §18.22(i) Permalink: http://dlmf.nist.gov/18.22.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

Meixner–Pollaczek

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

With

18.22.7		$p_{n}(x)=P^{(\lambda)}_{n}\left(x;\phi\right),$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : 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: TeX, pMML, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

18.22.8		$(n+1)p_{n+1}(x)=2\left(x\sin\phi+(n+\lambda)\cos\phi\right)p_{n}(x)-(n+2% \lambda-1)p_{n-1}(x).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(i), §18.30(v) Permalink: http://dlmf.nist.gov/18.22.E8 Encodings: TeX, pMML, png See also: Annotations for §18.22(i), §18.22(i), §18.22 and Ch.18

§18.22(ii) Difference Equations in $x$

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Keywords:: Hahn class orthogonal polynomials, difference equations on variable
Notes:: For (18.22.9)–(18.22.11) see Ismail (2009, (6.2.16)). For Table 18.22.2, Rows 2 and 3, see Ismail (2009, (6.2.38), (6.1.15)); Row 4 follows from Table 18.22.1, Row 4, and the third identity in (18.21.2). (18.22.13)–(18.22.15) is a limiting case of Koekoek et al. (2010, (9.1.6)) in view of (18.26.6). Koekoek et al. (2010, (9.1.6)) is a limiting case of Askey and Wilson (1985, (5.7)) in view of Koekoek et al. (2010, (14.1.21)). (18.22.16)–(18.22.17) follow from (18.22.14) in view of (18.21.10).
Permalink:: http://dlmf.nist.gov/18.22.ii
See also:: Annotations for §18.22 and Ch.18

Hahn

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

With

18.22.9		$p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right),$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E9 Encodings: TeX, pMML, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

18.22.10		$A(x)p_{n}(x+1)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-1)-n(n+\alpha+\beta+% 1)p_{n}(x)=0,$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable, $A(x)$ : coefficient and $C(x)$ : coefficient Referenced by: §16.4(iii) Permalink: http://dlmf.nist.gov/18.22.E10 Encodings: TeX, pMML, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

where

18.22.11	$\displaystyle A(x)$	$\displaystyle=(x+\alpha+1)(x-N),$
18.22.11	$\displaystyle C(x)$	$\displaystyle=x(x-\beta-N-1).$
ⓘ Defines: $A(x)$ : coefficient (locally) and $C(x)$ : coefficient (locally) Symbols: $N$ : positive integer and $x$ : real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

Krawtchouk, Meixner, and Charlier

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

18.22.12		$A(x)p_{n}(x+1)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-1)+\lambda_{n}p_{n}(% x)=0.$
ⓘ Symbols: $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable, $C(x)$ : coefficient, $\lambda_{n}$ : coefficient and $A(x)$ : coefficient Referenced by: §18.22(ii), Table 18.22.2 Permalink: http://dlmf.nist.gov/18.22.E12 Encodings: TeX, pMML, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

For $A(x)$ , $C(x)$ , and $\lambda_{n}$ in (18.22.12) see Table 18.22.2.

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

$p_{n}(x)$	$A(x)$	$C(x)$	$\lambda_{n}$
$K_{n}\left(x;p,N\right)$	$p(x-N)$	$(p-1)x$	$-n$
$M_{n}\left(x;\beta,c\right)$	$c(x+\beta)$	$x$	$n(1-c)$
$C_{n}\left(x;a\right)$	$a$	$x$	$n$

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Defines:: $C(x)$ : coefficient (locally), $\lambda_{n}$ : coefficient (locally) and $A(x)$ : coefficient (locally)
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, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(ii), §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.T2
See also:: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

Continuous Hahn

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

With

18.22.13		$p_{n}(x)=p_{n}\left(x;a,b,\overline{a},\overline{b}\right),$
ⓘ Symbols: $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn 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.E13 Encodings: TeX, pMML, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

18.22.14		$A(x)p_{n}(x+\mathrm{i})-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-\mathrm{i})% +n(n+2\Re\left(a+b\right)-1)p_{n}(x)=0,$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $\Re$ : real part, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable, $A(x)$ : coefficient and $C(x)$ : coefficient Referenced by: §18.22(ii), §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E14 Encodings: TeX, pMML, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

where

18.22.15	$\displaystyle A(x)$	$\displaystyle=(x+\mathrm{i}\overline{a})(x+\mathrm{i}\overline{b}),$
18.22.15	$\displaystyle C(x)$	$\displaystyle=(x-\mathrm{i}a)(x-\mathrm{i}b).$
ⓘ Defines: $A(x)$ : coefficient (locally) and $C(x)$ : coefficient (locally) Symbols: $\overline{\NVar{z}}$ : complex conjugate, $\mathrm{i}$ : imaginary unit and $x$ : real variable Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

Meixner–Pollaczek

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

With

18.22.16		$p_{n}(x)=P^{(\lambda)}_{n}\left(x;\phi\right),$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : 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: TeX, pMML, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

18.22.17		$A(x)p_{n}(x+\mathrm{i})-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-\mathrm{i})% +2n\sin\phi\,p_{n}(x)=0,$
ⓘ Symbols: $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $p_{n}(x)$ : polynomial of degree $n$ , $n$ : nonnegative integer, $x$ : real variable, $A(x)$ : coefficient and $C(x)$ : coefficient Referenced by: §18.22(ii) Permalink: http://dlmf.nist.gov/18.22.E17 Encodings: TeX, pMML, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

where

18.22.18	$\displaystyle A(x)$	$\displaystyle={\mathrm{e}}^{\mathrm{i}\phi}(x+\mathrm{i}\lambda),$
18.22.18	$\displaystyle C(x)$	$\displaystyle={\mathrm{e}}^{-\mathrm{i}\phi}(x-\mathrm{i}\lambda).$
ⓘ Defines: $A(x)$ : coefficient (locally) and $C(x)$ : coefficient (locally) Symbols: $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $x$ : real variable Permalink: http://dlmf.nist.gov/18.22.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §18.22(ii), §18.22(ii), §18.22 and Ch.18

§18.22(iii) $x$ -Differences

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Keywords:: Hahn class orthogonal polynomials, differences
Notes:: For (18.22.19)–(18.22.25) see Ismail (2009, (6.2.5), (6.2.13), (6.2.39), (6.2.40), (6.1.13), §6.1, unnumbered formula following (6.1.15), also (6.1.23)). (18.22.26) follows from (18.22.24) and (18.21.7). (18.22.27) follows from (18.20.9). (18.22.28) follows from (18.22.14) and (18.22.27). (18.22.29) follows from (18.20.10). (18.22.30) follows from (18.22.28) in view of (18.21.10).
Permalink:: http://dlmf.nist.gov/18.22.iii
See also:: Annotations for §18.22 and Ch.18

Hahn

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

18.22.19	$\displaystyle\Delta_{x}Q_{n}\left(x;\alpha,\beta,N\right)$	$\displaystyle=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}Q_{n-1}\left(x;\alpha+1,% \beta+1,N-1\right),$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, $\Delta$ : forward difference operator, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §16.4(iii), §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E19 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18
18.22.20	$\displaystyle\nabla_{x}\left(\frac{{\left(\alpha+1\right)_{x}}{\left(\beta+1% \right)_{N-x}}}{x!\;(N-x)!}Q_{n}\left(x;\alpha,\beta,N\right)\right)$	$\displaystyle=\frac{N+1}{\beta}\frac{{\left(\alpha\right)_{x}}{\left(\beta% \right)_{N+1-x}}}{x!\;(N+1-x)!}\*Q_{n+1}\left(x;\alpha-1,\beta-1,N+1\right).$
ⓘ Symbols: $Q_{\NVar{n}}\left(\NVar{x};\NVar{\alpha},\NVar{\beta},\NVar{N}\right)$ : Hahn polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\nabla_{\NVar{x}}$ : backward difference, $!$ : factorial (as in $n!$ ), $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable Referenced by: §16.4(iii) Permalink: http://dlmf.nist.gov/18.22.E20 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

Krawtchouk

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

18.22.21	$\displaystyle\Delta_{x}K_{n}\left(x;p,N\right)$	$\displaystyle=-\frac{n}{pN}K_{n-1}\left(x;p,N-1\right),$
ⓘ Symbols: $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $\Delta$ : forward difference operator, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.22.E21 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18
18.22.22	$\displaystyle\nabla_{x}\left(\genfrac{(}{)}{0.0pt}{}{N}{x}p^{x}(1-p)^{N-x}K_{n% }\left(x;p,N\right)\right)$	$\displaystyle=\genfrac{(}{)}{0.0pt}{}{N+1}{x}p^{x}{(1-p)^{N-x}}K_{n+1}\left(x;% p,N+1\right).$
ⓘ Symbols: $K_{\NVar{n}}\left(\NVar{x};\NVar{p},\NVar{N}\right)$ : Krawtchouk polynomial, $\nabla_{\NVar{x}}$ : backward difference, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $N$ : positive integer, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.22.E22 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

Meixner

ⓘ

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

18.22.23		$\Delta_{x}M_{n}\left(x;\beta,c\right)=-\frac{n(1-c)}{\beta c}M_{n-1}\left(x;% \beta+1,c\right),$
ⓘ Symbols: $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, $\Delta$ : forward difference operator, $n$ : nonnegative integer and $x$ : real variable Permalink: http://dlmf.nist.gov/18.22.E23 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

18.22.24		$\nabla_{x}\left(\frac{{\left(\beta\right)_{x}}c^{x}}{x!}M_{n}\left(x;\beta,c% \right)\right)=\frac{{\left(\beta-1\right)_{x}}c^{x}}{x!}M_{n+1}\left(x;\beta-% 1,c\right).$
ⓘ Symbols: $M_{\NVar{n}}\left(\NVar{x};\NVar{\beta},\NVar{c}\right)$ : Meixner polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\nabla_{\NVar{x}}$ : backward difference, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E24 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

Charlier

ⓘ

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

18.22.25	$\displaystyle\Delta_{x}C_{n}\left(x;a\right)$	$\displaystyle=-\frac{n}{a}C_{n-1}\left(x;a\right),$
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\Delta$ : forward difference operator, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E25 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18
18.22.26	$\displaystyle\nabla_{x}\left(\frac{a^{x}}{x!}C_{n}\left(x;a\right)\right)$	$\displaystyle=\frac{a^{x}}{x!}C_{n+1}\left(x;a\right).$
ⓘ Symbols: $C_{\NVar{n}}\left(\NVar{x};\NVar{a}\right)$ : Charlier polynomial, $\nabla_{\NVar{x}}$ : backward difference, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E26 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

Continuous Hahn

ⓘ

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

18.22.27		$\delta_{x}\left(p_{n}\left(x;a,b,\overline{a},\overline{b}\right)\right)=(n+2% \Re\left(a+b\right)-1)\*p_{n-1}\left(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline% {a}+\tfrac{1}{2},\overline{b}+\tfrac{1}{2}\right),$
ⓘ Symbols: $\delta_{\NVar{x}}$ : central difference, $\overline{\NVar{z}}$ : complex conjugate, $p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$ : continuous Hahn polynomial, $\Re$ : real part, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E27 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

18.22.28		$\delta_{x}\left(w(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+\tfrac{1}{2},% \overline{b}+\tfrac{1}{2})p_{n}(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\overline{a}+% \tfrac{1}{2},\overline{b}+\tfrac{1}{2})\right)=-(n+1)w(x;a,b,\overline{a},% \overline{b})p_{n+1}(x;a,b,\overline{a},\overline{b}).$
ⓘ Symbols: $\delta_{\NVar{x}}$ : central difference, $\overline{\NVar{z}}$ : complex conjugate, $p_{n}(x)$ : polynomial of degree $n$ , $w(x)$ : weight function, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.20(i), §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E28 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

Meixner–Pollaczek

ⓘ

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

18.22.29		$\delta_{x}\left(P^{(\lambda)}_{n}\left(x;\phi\right)\right)=2\sin\phi P^{(% \lambda+\frac{1}{2})}_{n-1}\left(x;\phi\right),$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $\delta_{\NVar{x}}$ : central difference, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $x$ : real variable Referenced by: §18.22(iii) Permalink: http://dlmf.nist.gov/18.22.E29 Encodings: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

18.22.30		$\delta_{x}\left(w^{(\lambda+\frac{1}{2})}(x;\phi)P^{(\lambda+\frac{1}{2})}_{n}% \left(x;\phi\right)\right)=-(n+1)w^{(\lambda)}(x;\phi)P^{(\lambda)}_{n+1}\left% (x;\phi\right).$
ⓘ Symbols: $P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$ : Meixner–Pollaczek polynomial, $\delta_{\NVar{x}}$ : 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: TeX, pMML, png See also: Annotations for §18.22(iii), §18.22(iii), §18.22 and Ch.18

18.21 Hahn Class: Interrelations 18.23 Hahn Class: Generating Functions

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§18.22 Hahn Class: Recurrence Relations and Differences

Contents

§18.22(i) Recurrence Relations in $n$

Hahn

Krawtchouk, Meixner, and Charlier

Continuous Hahn

Meixner–Pollaczek

§18.22(ii) Difference Equations in $x$

Hahn

Krawtchouk, Meixner, and Charlier

Continuous Hahn

Meixner–Pollaczek

§18.22(iii) $x$ -Differences

Hahn

Krawtchouk

Meixner

Charlier

Continuous Hahn

Meixner–Pollaczek

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

§18.22 Hahn Class: Recurrence Relations and Differences

Contents

§18.22(i) Recurrence Relations in n

Hahn

Krawtchouk, Meixner, and Charlier

Continuous Hahn

Meixner–Pollaczek

§18.22(ii) Difference Equations in x

Hahn

Krawtchouk, Meixner, and Charlier

Continuous Hahn

Meixner–Pollaczek

§18.22(iii) x-Differences

Hahn

Krawtchouk

Meixner

Charlier

Continuous Hahn

Meixner–Pollaczek

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!

§18.22(i) Recurrence Relations in $n$

§18.22(ii) Difference Equations in $x$

§18.22(iii) $x$ -Differences