§18.22 Hahn Class: Recurrence Relations and Differences

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

§18.22(i) Recurrence Relations in $n$
§18.22(ii) Difference Equations in $x$
§18.22(iii) $x$ -Differences

§18.22(i) Recurrence Relations in $n$

Notes:: For (18.22.1)–(18.22.3) see Ismail (2005, (6.2.8) and (6.2.9)). For Table 18.22.1 see Ismail (2005, (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 (2005, (5.9.1)).
Keywords:: Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.22.i

¶ Hahn

With

18.22.1 $p_{n}(x)=\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right),$

Symbols:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn polynomial, $n$ : nonnegative integer, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.22(i)
Permalink:: http://dlmf.nist.gov/18.22.E1
Encodings:: TeX, pMML, png

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:: $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : 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

where

18.22.3

$A_{n}=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)},$

$C_{n}=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}.$

Symbols:: $n$ : nonnegative integer, $N$ : positive integer, $A_{n}$ : coefficient and $C_{n}$ : coefficient
Referenced by:: §18.22(i)
Permalink:: http://dlmf.nist.gov/18.22.E3
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¶ Krawtchouk, Meixner, and Charlier

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.

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

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, $n$ : nonnegative integer, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $A_{n}$ : coefficient and $C_{n}$ : coefficient
Referenced by:: §18.21(ii), ¶ ‣ §18.22(i), §18.22(i), §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.T1

¶ Continuous Hahn

With

18.22.4 $q_{n}(x)=\ifrac{\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)}{\mathop{p_{{n}}\/}\nolimits\!\left(ia;a,b,\conj{a},\conj{b}\right)},$

Symbols:: $\conj{z}$ : complex conjugate, $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(i)
Permalink:: http://dlmf.nist.gov/18.22.E4
Encodings:: TeX, pMML, png

18.22.5 $(a+ix)q_{{n}}(x)=\tilde{A}_{n}q_{{n+1}}(x)-\bigl(\tilde{A}_{n}+\tilde{C}_{n}\bigr)q_{n}(x)+\tilde{C}_{n}q_{{n-1}}(x),$

Symbols:: $n$ : 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

where

18.22.6

$\tilde{A}_{n}=-\frac{(n+2\realpart{(a+b)}-1)(n+a+\conj{a})(n+a+\conj{b})}{(2n+2\realpart{(a+b)}-1)(2n+2\realpart{(a+b)})},$

$\tilde{C}_{n}=\frac{n(n+b+\conj{a}-1)(n+b+\conj{b}-1)}{(2n+2\realpart{(a+b)}-2)(2n+2\realpart{(a+b)}-1)}.$

Symbols:: $\conj{z}$ : complex conjugate, $\realpart{}$ : real part, $n$ : nonnegative integer, $\tilde{A}_{n}$ : coefficient and $\tilde{C}_{n}$ : coefficient
Referenced by:: §18.22(i)
Permalink:: http://dlmf.nist.gov/18.22.E6
Encodings:: TeX, TeX, pMML, pMML, png, png

¶ Meixner–Pollaczek

With

18.22.7 $p_{n}(x)=\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right),$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.22(i)
Permalink:: http://dlmf.nist.gov/18.22.E7
Encodings:: TeX, pMML, png

18.22.8 $(n+1)p_{{n+1}}(x)=2\left(x\mathop{\sin\/}\nolimits\phi+(n+\lambda)\mathop{\cos\/}\nolimits\phi\right)p_{n}(x)-(n+2\lambda-1)p_{{n-1}}(x).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.22(i)
Permalink:: http://dlmf.nist.gov/18.22.E8
Encodings:: TeX, pMML, png

§18.22(ii) Difference Equations in $x$

Notes:: For (18.22.9)–(18.22.11) see Ismail (2005, (6.2.16)). For Table 18.22.2, Rows 2 and 3, see Ismail (2005, (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, (1.1.6)) in view of (18.26.6). Koekoek et al. (2010, (1.1.6)) is a limiting case of Askey and Wilson (1985, (5.7)) in view of Koekoek et al. (2010, (5.1.1)). (18.22.16)–(18.22.17) follow from (18.22.14) in view of (18.21.10).
Keywords:: Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.22.ii

¶ Hahn

With

18.22.9 $p_{n}(x)=\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right),$

Symbols:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn polynomial, $n$ : nonnegative integer, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.E9
Encodings:: TeX, pMML, png

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:: $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : 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

where

18.22.11

$A(x)=(x+\alpha+1)(x-N),$

$C(x)=x(x-\beta-N-1).$

Symbols:: $N$ : positive integer, $x$ : real variable, $A(x)$ : coefficient and $C(x)$ : coefficient
Referenced by:: §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

¶ Krawtchouk, Meixner, and Charlier

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:: $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : 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

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.

$p_{n}(x)$	$A(x)$	$C(x)$	$\lambda _{n}$
$\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$	$p(x-N)$	$(p-1)x$	$-n$
$\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$	$c(x+\beta)$	$x$	$n(1-c)$
$\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$	$a$	$x$	$n$

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, $n$ : nonnegative integer, $N$ : positive integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : real variable, $C(x)$ : coefficient, $\lambda _{n}$ : coefficient and $A(x)$ : coefficient
Referenced by:: ¶ ‣ §18.22(ii), §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.T2

¶ Continuous Hahn

With

18.22.13 $p_{n}(x)=\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right),$

Symbols:: $\conj{z}$ : complex conjugate, $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.E13
Encodings:: TeX, pMML, png

18.22.14 $A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+n(n+2\realpart{(a+b)}-1)p_{n}(x)=0,$

Symbols:: $\realpart{}$ : real part, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : 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

where

18.22.15

$A(x)=(x+i\conj{a})(x+i\conj{b}),$

$C(x)=(x-ia)(x-ib).$

Symbols:: $\conj{z}$ : complex conjugate, $x$ : real variable, $A(x)$ : coefficient and $C(x)$ : coefficient
Referenced by:: §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.E15
Encodings:: TeX, TeX, pMML, pMML, png, png

¶ Meixner–Pollaczek

With

18.22.16 $p_{n}(x)=\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right),$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.22(ii)
Permalink:: http://dlmf.nist.gov/18.22.E16
Encodings:: TeX, pMML, png

18.22.17 $A(x)p_{n}(x+i)-\left(A(x)+C(x)\right)p_{n}(x)+C(x)p_{n}(x-i)+2n\mathop{\sin\/}\nolimits\phi\, p_{n}(x)=0,$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ , $x$ : 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

where

18.22.18

$A(x)=e^{{i\phi}}(x+i\lambda),$

$C(x)=e^{{-i\phi}}(x-i\lambda).$

Symbols:: $e$ : base of exponential function, $x$ : real variable, $A(x)$ : coefficient and $C(x)$ : coefficient
Permalink:: http://dlmf.nist.gov/18.22.E18
Encodings:: TeX, TeX, pMML, pMML, png, png

§18.22(iii) $x$ -Differences

Notes:: For (18.22.19)–(18.22.25) see Ismail (2005, (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).
Keywords:: Hahn class orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.22.iii

¶ Hahn

18.22.19 $\Delta _{{x}}\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}\mathop{Q_{{n-1}}\/}\nolimits\!\left(x;\alpha+1,\beta+1,N-1\right),$

Symbols:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn polynomial, $\Delta$ : forward difference operator, $n$ : nonnegative integer, $N$ : positive 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

18.22.20 $\nabla _{{x}}\left(\frac{\left(\alpha+1\right)_{{x}}\left(\beta+1\right)_{{N-x}}}{x!\;(N-x)!}\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)\right)=\frac{N+1}{\beta}\frac{\left(\alpha\right)_{{x}}\left(\beta\right)_{{N+1-x}}}{x!\;(N+1-x)!}\*\mathop{Q_{{n+1}}\/}\nolimits\!\left(x;\alpha-1,\beta-1,N+1\right).$

Symbols:: $\mathop{Q_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,N\right)$ : Hahn polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\nabla _{{x}}$ : backward difference, $!$ : $n!$ : factorial, $n$ : nonnegative integer, $N$ : positive integer and $x$ : real variable
Referenced by:: §16.4(iii)
Permalink:: http://dlmf.nist.gov/18.22.E20
Encodings:: TeX, pMML, png

¶ Krawtchouk

18.22.21 $\Delta _{{x}}\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)=-\frac{n}{pN}\mathop{K_{{n-1}}\/}\nolimits\!\left(x;p,N-1\right),$

Symbols:: $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, $\Delta$ : forward difference operator, $n$ : nonnegative integer, $N$ : positive integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.22.E21
Encodings:: TeX, pMML, png

18.22.22 $\nabla _{{x}}\left(\binom{N}{x}p^{x}(1-p)^{{N-x}}\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)\right)=\binom{N+1}{x}p^{x}{(1-p)^{{N-x}}}\mathop{K_{{n+1}}\/}\nolimits\!\left(x;p,N+1\right).$

Symbols:: $\mathop{K_{{n}}\/}\nolimits\!\left(x;p,N\right)$ : Krawtchouk polynomial, $\nabla _{{x}}$ : backward difference, $\binom{m}{n}$ : binomial coefficient, $n$ : nonnegative integer, $N$ : positive integer and $x$ : real variable
Permalink:: http://dlmf.nist.gov/18.22.E22
Encodings:: TeX, pMML, png

¶ Meixner

18.22.23 $\Delta _{{x}}\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)=-\frac{n(1-c)}{\beta c}\mathop{M_{{n-1}}\/}\nolimits\!\left(x;\beta+1,c\right),$

Symbols:: $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : 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

18.22.24 $\nabla _{{x}}\left(\frac{\left(\beta\right)_{{x}}c^{x}}{x!}\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)\right)=\frac{\left(\beta-1\right)_{{x}}c^{x}}{x!}\mathop{M_{{n+1}}\/}\nolimits\!\left(x;\beta-1,c\right).$

Symbols:: $\mathop{M_{{n}}\/}\nolimits\!\left(x;\beta,c\right)$ : Meixner polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\nabla _{{x}}$ : backward difference, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.22.E24
Encodings:: TeX, pMML, png

¶ Charlier

18.22.25 $\Delta _{{x}}\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)=-\frac{n}{a}\mathop{C_{{n-1}}\/}\nolimits\!\left(x,a\right),$

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : 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

18.22.26 $\nabla _{{x}}\left(\frac{a^{x}}{x!}\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)\right)=\frac{a^{x}}{x!}\mathop{C_{{n+1}}\/}\nolimits\!\left(x,a\right).$

Symbols:: $\mathop{C_{{n}}\/}\nolimits\!\left(x,a\right)$ : Charlier polynomial, $\nabla _{{x}}$ : backward difference, $!$ : $n!$ : factorial, $n$ : nonnegative integer and $x$ : real variable
Referenced by:: §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.22.E26
Encodings:: TeX, pMML, png

¶ Continuous Hahn

18.22.27 $\delta _{{x}}\left(\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)\right)=(n+2\realpart{(a+b)}-1)\*\mathop{p_{{n-1}}\/}\nolimits\!\left(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\conj{a}+\tfrac{1}{2},\conj{b}+\tfrac{1}{2}\right),$

Symbols:: $\delta _{{x}}$ : central difference, $\conj{z}$ : complex conjugate, $\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$ : continuous Hahn polynomial, $\realpart{}$ : 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

18.22.28 $\delta _{{x}}\left(w(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\conj{a}+\tfrac{1}{2},\conj{b}+\tfrac{1}{2})p_{n}(x;a+\tfrac{1}{2},b+\tfrac{1}{2},\conj{a}+\tfrac{1}{2},\conj{b}+\tfrac{1}{2})\right)=-(n+1)w(x;a,b,\conj{a},\conj{b})p_{{n+1}}(x;a,b,\conj{a},\conj{b}).$

Symbols:: $\delta _{{x}}$ : central difference, $\conj{z}$ : complex conjugate, $w(x)$ : weight function, $n$ : nonnegative integer, $p_{n}(x)$ : polynomial of degree $n$ and $x$ : real variable
Referenced by:: §18.20(i), §18.22(iii)
Permalink:: http://dlmf.nist.gov/18.22.E28
Encodings:: TeX, pMML, png

¶ Meixner–Pollaczek

18.22.29 $\delta _{{x}}\left(\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)\right)=2\mathop{\sin\/}\nolimits\phi\mathop{P^{{(\lambda+\frac{1}{2})}}_{{n-1}}\/}\nolimits\!\left(x;\phi\right),$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : Meixner–Pollaczek polynomial, $\delta _{{x}}$ : central difference, $\mathop{\sin\/}\nolimits z$ : 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

18.22.30 $\delta _{{x}}\left(w^{{(\lambda+\frac{1}{2})}}(x;\phi)\mathop{P^{{(\lambda+\frac{1}{2})}}_{{n}}\/}\nolimits\!\left(x;\phi\right)\right)=-(n+1)w^{{(\lambda)}}(x;\phi)\mathop{P^{{(\lambda)}}_{{n+1}}\/}\nolimits\!\left(x;\phi\right).$

Symbols:: $\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$ : 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:: TeX, pMML, png

§18.22 Hahn Class: Recurrence Relations and Differences

Contents

§18.22(i) Recurrence Relations in

¶ Hahn

¶ Krawtchouk, Meixner, and Charlier

¶ Continuous Hahn

¶ Meixner–Pollaczek

§18.22(ii) Difference Equations in

¶ Hahn

¶ Krawtchouk, Meixner, and Charlier

¶ Continuous Hahn

¶ Meixner–Pollaczek

§18.22(iii) -Differences

¶ Hahn

¶ Krawtchouk

¶ Meixner

¶ Charlier

¶ Continuous Hahn

¶ Meixner–Pollaczek

§18.22(i) Recurrence Relations in $n$

§18.22(ii) Difference Equations in $x$

§18.22(iii) $x$ -Differences