Meixner

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Meixner

Hypergeometric representation

M n ⁡ ( x ; β , c ) = \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , - x ⁢ β ⁢ 1 - 1 c Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 \HyperpFq 21 @ @ 𝑛 𝑥 𝛽 1 1 𝑐 {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(x;\beta,c\right)=% \HyperpFq{2}{1}@@{-n,-x}{\beta}{1-\frac{1}{c}}}}} {\displaystyle \Meixner{n}@{x}{\beta}{c}=\HyperpFq{2}{1}@@{-n,-x}{\beta}{1-\frac{1}{c}} }

Orthogonality relation(s)

∑ x = 0 ∞ ( β ) x x ! ⁢ c x ⁢ M m ⁡ ( x ; β , c ) ⁢ M n ⁡ ( x ; β , c ) = c - n ⁢ n ! ( β ) n ⁢ ( 1 - c ) β ⁢ δ m , n superscript subscript 𝑥 0 Pochhammer-symbol 𝛽 𝑥 𝑥 superscript 𝑐 𝑥 Meixner-polynomial-M 𝑚 𝑥 𝛽 𝑐 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 superscript 𝑐 𝑛 𝑛 Pochhammer-symbol 𝛽 𝑛 superscript 1 𝑐 𝛽 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{{\left(% \beta\right)_{x}}}{x!}c^{x}M_{m}\!\left(x;\beta,c\right)M_{n}\!\left(x;\beta,c% \right){}=\frac{c^{-n}n!}{{\left(\beta\right)_{n}}(1-c)^{\beta}}\,\delta_{m,n}% }}} {\displaystyle \sum_{x=0}^{\infty}\frac{\pochhammer{\beta}{x}}{x!}c^x\Meixner{m}@{x}{\beta}{c}\Meixner{n}@{x}{\beta}{c} {}=\frac{c^{-n}n!}{\pochhammer{\beta}{n}(1-c)^{\beta}}\,\Kronecker{m}{n} }

Constraint(s): β > 0   0 < c < 1 formulae-sequence 𝛽 0 0 𝑐 1 {\displaystyle{\displaystyle{\displaystyle\beta>0\quad 0<c<1}}}


Recurrence relation

( c - 1 ) ⁢ x ⁢ M n ⁡ ( x ; β , c ) = c ⁢ ( n + β ) ⁢ M n + 1 ⁡ ( x ; β , c ) - [ n + ( n + β ) ⁢ c ] ⁢ M n ⁡ ( x ; β , c ) + n ⁢ M n - 1 ⁡ ( x ; β , c ) 𝑐 1 𝑥 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 𝑐 𝑛 𝛽 Meixner-polynomial-M 𝑛 1 𝑥 𝛽 𝑐 delimited-[] 𝑛 𝑛 𝛽 𝑐 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 𝑛 Meixner-polynomial-M 𝑛 1 𝑥 𝛽 𝑐 {\displaystyle{\displaystyle{\displaystyle(c-1)xM_{n}\!\left(x;\beta,c\right)=% c(n+\beta)M_{n+1}\!\left(x;\beta,c\right){}-\left[n+(n+\beta)c\right]M_{n}\!% \left(x;\beta,c\right)+nM_{n-1}\!\left(x;\beta,c\right)}}} {\displaystyle (c-1)x\Meixner{n}@{x}{\beta}{c}=c(n+\beta)\Meixner{n+1}@{x}{\beta}{c} {}-\left[n+(n+\beta)c\right]\Meixner{n}@{x}{\beta}{c}+n\Meixner{n-1}@{x}{\beta}{c} }

Monic recurrence relation

x ⁢ M ^ n ⁡ ( x ; β , c ) = M ^ n + 1 ⁡ ( x ; β , c ) + n + ( n + β ) ⁢ c 1 - c ⁢ M ^ n ⁡ ( x ; β , c ) + n ⁢ ( n + β - 1 ) ⁢ c ( 1 - c ) 2 ⁢ M ^ n - 1 ⁡ ( x ; β , c ) 𝑥 Meixner-polynomial-monic-p 𝑛 𝑥 𝛽 𝑐 Meixner-polynomial-monic-p 𝑛 1 𝑥 𝛽 𝑐 𝑛 𝑛 𝛽 𝑐 1 𝑐 Meixner-polynomial-monic-p 𝑛 𝑥 𝛽 𝑐 𝑛 𝑛 𝛽 1 𝑐 superscript 1 𝑐 2 Meixner-polynomial-monic-p 𝑛 1 𝑥 𝛽 𝑐 {\displaystyle{\displaystyle{\displaystyle x{\widehat{M}}_{n}\!\left(x;\beta,c% \right)={\widehat{M}}_{n+1}\!\left(x;\beta,c\right)+\frac{n+(n+\beta)c}{1-c}{% \widehat{M}}_{n}\!\left(x;\beta,c\right)+\frac{n(n+\beta-1)c}{(1-c)^{2}}{% \widehat{M}}_{n-1}\!\left(x;\beta,c\right)}}} {\displaystyle x\monicMeixner{n}@@{x}{\beta}{c}=\monicMeixner{n+1}@@{x}{\beta}{c}+\frac{n+(n+\beta)c}{1-c}\monicMeixner{n}@@{x}{\beta}{c}+ \frac{n(n+\beta-1)c}{(1-c)^2}\monicMeixner{n-1}@@{x}{\beta}{c} }
M n ⁡ ( x ; β , c ) = 1 ( β ) n ⁢ ( c - 1 c ) n ⁢ M ^ n ⁡ ( x ; β , c ) Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 1 Pochhammer-symbol 𝛽 𝑛 superscript 𝑐 1 𝑐 𝑛 Meixner-polynomial-monic-p 𝑛 𝑥 𝛽 𝑐 {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(x;\beta,c\right)=\frac% {1}{{\left(\beta\right)_{n}}}\left(\frac{c-1}{c}\right)^{n}{\widehat{M}}_{n}\!% \left(x;\beta,c\right)}}} {\displaystyle \Meixner{n}@{x}{\beta}{c}=\frac{1}{\pochhammer{\beta}{n}}\left(\frac{c-1}{c}\right)^n\monicMeixner{n}@@{x}{\beta}{c} }

Difference equation

n ⁢ ( c - 1 ) ⁢ y ⁢ ( x ) = c ⁢ ( x + β ) ⁢ y ⁢ ( x + 1 ) - [ x + ( x + β ) ⁢ c ] ⁢ y ⁢ ( x ) + x ⁢ y ⁢ ( x - 1 ) 𝑛 𝑐 1 𝑦 𝑥 𝑐 𝑥 𝛽 𝑦 𝑥 1 delimited-[] 𝑥 𝑥 𝛽 𝑐 𝑦 𝑥 𝑥 𝑦 𝑥 1 {\displaystyle{\displaystyle{\displaystyle n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x% +(x+\beta)c\right]y(x)+xy(x-1)}}} {\displaystyle n(c-1)y(x)=c(x+\beta)y(x+1)-\left[x+(x+\beta)c\right]y(x)+xy(x-1) }

Substitution(s): y ⁢ ( x ) = M n ⁡ ( x ; β , c ) 𝑦 𝑥 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 {\displaystyle{\displaystyle{\displaystyle y(x)=M_{n}\!\left(x;\beta,c\right)}}}


Forward shift operator

M n ⁡ ( x + 1 ; β , c ) - M n ⁡ ( x ; β , c ) = n β ⁢ ( c - 1 c ) ⁢ M n - 1 ⁡ ( x ; β + 1 , c ) Meixner-polynomial-M 𝑛 𝑥 1 𝛽 𝑐 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 𝑛 𝛽 𝑐 1 𝑐 Meixner-polynomial-M 𝑛 1 𝑥 𝛽 1 𝑐 {\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(x+1;\beta,c\right)-M_{% n}\!\left(x;\beta,c\right)=\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}\!% \left(x;\beta+1,c\right)}}} {\displaystyle \Meixner{n}@{x+1}{\beta}{c}-\Meixner{n}@{x}{\beta}{c}= \frac{n}{\beta}\left(\frac{c-1}{c}\right)\Meixner{n-1}@{x}{\beta+1}{c} }
Δ ⁢ M n ⁡ ( x ; β , c ) = n β ⁢ ( c - 1 c ) ⁢ M n - 1 ⁡ ( x ; β + 1 , c ) Δ Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 𝑛 𝛽 𝑐 1 𝑐 Meixner-polynomial-M 𝑛 1 𝑥 𝛽 1 𝑐 {\displaystyle{\displaystyle{\displaystyle\Delta M_{n}\!\left(x;\beta,c\right)% =\frac{n}{\beta}\left(\frac{c-1}{c}\right)M_{n-1}\!\left(x;\beta+1,c\right)}}} {\displaystyle \Delta \Meixner{n}@{x}{\beta}{c}=\frac{n}{\beta}\left(\frac{c-1}{c}\right)\Meixner{n-1}@{x}{\beta+1}{c} }

Backward shift operator

c ⁢ ( β + x - 1 ) ⁢ M n ⁡ ( x ; β , c ) - x ⁢ M n ⁡ ( x - 1 ; β , c ) = c ⁢ ( β - 1 ) ⁢ M n + 1 ⁡ ( x ; β - 1 , c ) 𝑐 𝛽 𝑥 1 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 𝑥 Meixner-polynomial-M 𝑛 𝑥 1 𝛽 𝑐 𝑐 𝛽 1 Meixner-polynomial-M 𝑛 1 𝑥 𝛽 1 𝑐 {\displaystyle{\displaystyle{\displaystyle c(\beta+x-1)M_{n}\!\left(x;\beta,c% \right)-xM_{n}\!\left(x-1;\beta,c\right)=c(\beta-1)M_{n+1}\!\left(x;\beta-1,c% \right)}}} {\displaystyle c(\beta+x-1)\Meixner{n}@{x}{\beta}{c}-x\Meixner{n}@{x-1}{\beta}{c}=c(\beta-1)\Meixner{n+1}@{x}{\beta-1}{c} }
∇ ⁡ [ ( β ) x ⁢ c x x ! ⁢ M n ⁡ ( x ; β , c ) ] = ( β - 1 ) x ⁢ c x x ! ⁢ M n + 1 ⁡ ( x ; β - 1 , c ) ∇ Pochhammer-symbol 𝛽 𝑥 superscript 𝑐 𝑥 𝑥 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 Pochhammer-symbol 𝛽 1 𝑥 superscript 𝑐 𝑥 𝑥 Meixner-polynomial-M 𝑛 1 𝑥 𝛽 1 𝑐 {\displaystyle{\displaystyle{\displaystyle\nabla\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)}}} {\displaystyle \nabla\left[\frac{\pochhammer{\beta}{x}c^x}{x!}\Meixner{n}@{x}{\beta}{c}\right]= \frac{\pochhammer{\beta-1}{x}c^x}{x!}\Meixner{n+1}@{x}{\beta-1}{c} }

Rodrigues-type formula

( β ) x ⁢ c x x ! ⁢ M n ⁡ ( x ; β , c ) = ∇ n ⁡ [ ( β + n ) x ⁢ c x x ! ] Pochhammer-symbol 𝛽 𝑥 superscript 𝑐 𝑥 𝑥 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 superscript ∇ 𝑛 Pochhammer-symbol 𝛽 𝑛 𝑥 superscript 𝑐 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{{\left(\beta\right)_{x}}c^{x}}% {x!}M_{n}\!\left(x;\beta,c\right)=\nabla^{n}\left[\frac{{\left(\beta+n\right)_% {x}}c^{x}}{x!}\right]}}} {\displaystyle \frac{\pochhammer{\beta}{x}c^x}{x!}\Meixner{n}@{x}{\beta}{c}=\nabla^n\left[\frac{\pochhammer{\beta+n}{x}c^x}{x!}\right] }

Generating functions

( 1 - t c ) x ⁢ ( 1 - t ) - x - β = ∑ n = 0 ∞ ( β ) n n ! ⁢ M n ⁡ ( x ; β , c ) ⁢ t n superscript 1 𝑡 𝑐 𝑥 superscript 1 𝑡 𝑥 𝛽 superscript subscript 𝑛 0 Pochhammer-symbol 𝛽 𝑛 𝑛 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(1-\frac{t}{c}\right)^{x}(1-t)^% {-x-\beta}=\sum_{n=0}^{\infty}\frac{{\left(\beta\right)_{n}}}{n!}M_{n}\!\left(% x;\beta,c\right)t^{n}}}} {\displaystyle \left(1-\frac{t}{c}\right)^x(1-t)^{-x-\beta}= \sum_{n=0}^{\infty}\frac{\pochhammer{\beta}{n}}{n!}\Meixner{n}@{x}{\beta}{c}t^n }
e t ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ - x ⁢ β ⁢ ( 1 - c c ) ⁢ t = ∑ n = 0 ∞ M n ⁡ ( x ; β , c ) n ! ⁢ t n 𝑡 \HyperpFq 11 @ @ 𝑥 𝛽 1 𝑐 𝑐 𝑡 superscript subscript 𝑛 0 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{1}{1}@@{% -x}{\beta}{\left(\frac{1-c}{c}\right)t}=\sum_{n=0}^{\infty}\frac{M_{n}\!\left(% x;\beta,c\right)}{n!}t^{n}}}} {\displaystyle \expe^t\,\HyperpFq{1}{1}@@{-x}{\beta}{\left(\frac{1-c}{c}\right)t}= \sum_{n=0}^{\infty}\frac{\Meixner{n}@{x}{\beta}{c}}{n!}t^n }
( 1 - t ) - γ ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , - x ⁢ β ⁢ ( 1 - c ) ⁢ t c ⁢ ( 1 - t ) = ∑ n = 0 ∞ ( γ ) n n ! ⁢ M n ⁡ ( x ; β , c ) ⁢ t n superscript 1 𝑡 𝛾 \HyperpFq 21 @ @ 𝛾 𝑥 𝛽 1 𝑐 𝑡 𝑐 1 𝑡 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 𝑛 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{% \gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}}=\sum_{n=0}^{\infty}\frac{{\left(% \gamma\right)_{n}}}{n!}M_{n}\!\left(x;\beta,c\right)t^{n}}}} {\displaystyle (1-t)^{-\gamma}\,\HyperpFq{2}{1}@@{\gamma,-x}{\beta}{\frac{(1-c)t}{c(1-t)}} =\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{n!}\Meixner{n}@{x}{\beta}{c}t^n }

Constraint(s): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Hahn polynomial to Meixner polynomial

lim N → ∞ ⁡ Q n ⁡ ( x ; b - 1 , N ⁢ ( 1 - c ) ⁢ c - 1 , N ) = M n ⁡ ( x ; b , c ) subscript → 𝑁 Hahn-polynomial-Q 𝑛 𝑥 𝑏 1 𝑁 1 𝑐 superscript 𝑐 1 𝑁 Meixner-polynomial-M 𝑛 𝑥 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!% \left(x;b-1,N(1-c)c^{-1},N\right)=M_{n}\!\left(x;b,c\right)}}} {\displaystyle \lim_{N\rightarrow\infty} \Hahn{n}@{x}{b-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{b}{c} }

Dual Hahn polynomial to Meixner polynomial

lim N → ∞ ⁡ R n ⁡ ( λ ⁢ ( x ) ; β - 1 , N ⁢ ( 1 - c ) ⁢ c - 1 , N ) = M n ⁡ ( x ; β , c ) subscript → 𝑁 dual-Hahn-R 𝑛 𝜆 𝑥 𝛽 1 𝑁 1 𝑐 superscript 𝑐 1 𝑁 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}R_{n}\!% \left(\lambda(x);\beta-1,N(1-c)c^{-1},N\right)=M_{n}\!\left(x;\beta,c\right)}}} {\displaystyle \lim_{N\rightarrow\infty} \dualHahn{n}@{\lambda(x)}{\beta-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{\beta}{c} }

Meixner polynomial to Laguerre polynomial

lim c → 1 ⁡ M n ⁡ ( ( 1 - c ) - 1 ⁢ x ; α + 1 , c ) = L n α ⁡ ( x ) L n α ⁡ ( 0 ) subscript → 𝑐 1 Meixner-polynomial-M 𝑛 superscript 1 𝑐 1 𝑥 𝛼 1 𝑐 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 0 {\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow 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)}}}} {\displaystyle \lim_{c\rightarrow 1} \Meixner{n}@{(1-c)^{-1}x}{\alpha+1}{c}=\frac{\Laguerre[\alpha]{n}@{x}}{\Laguerre[\alpha]{n}@{0}} }

Meixner polynomial to Charlier polynomial

lim β → ∞ ⁡ M n ⁡ ( x ; β , ( a + β ) - 1 ⁢ a ) = C n ⁡ ( x ; a ) subscript → 𝛽 Meixner-polynomial-M 𝑛 𝑥 𝛽 superscript 𝑎 𝛽 1 𝑎 Charlier-polynomial-C 𝑛 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}M_{n}\!% \left(x;\beta,(a+\beta)^{-1}a\right)=C_{n}\!\left(x;a\right)}}} {\displaystyle \lim_{\beta\rightarrow\infty} \Meixner{n}@{x}{\beta}{(a+\beta)^{-1}a}=\Charlier{n}@{x}{a} }

Remarks

( β ) n n ! M n ⁡ ( x ; β , c ) = P n ( β - 1 , - n - β - x ) ⁡ ( ( 2 - c ) c - 1 ) fragments Pochhammer-symbol 𝛽 𝑛 𝑛 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 Jacobi-polynomial-P 𝛽 1 𝑛 𝛽 𝑥 𝑛 fragments ( 2 c superscript 𝑐 1 ) {\displaystyle{\displaystyle{\displaystyle\frac{{\left(\beta\right)_{n}}}{n!}M% _{n}\!\left(x;\beta,c\right)=P^{(\beta-1,-n-\beta-x)}_{n}\left((2-c\right)c^{-% 1})}}} {\displaystyle \frac{\pochhammer{\beta}{n}}{n!}\Meixner{n}@{x}{\beta}{c}=\Jacobi{\beta-1}{-n-\beta-x}{n}@{(2-c}c^{-1}) }
K n ⁡ ( x ; p , N ) = M n ⁡ ( x ; - N , ( p - 1 ) - 1 ⁢ p ) Krawtchouk-polynomial-K 𝑛 𝑥 𝑝 𝑁 Meixner-polynomial-M 𝑛 𝑥 𝑁 superscript 𝑝 1 1 𝑝 {\displaystyle{\displaystyle{\displaystyle K_{n}\!\left(x;p,N\right)=M_{n}\!% \left(x;-N,(p-1)^{-1}p\right)}}} {\displaystyle \Krawtchouk{n}@{x}{p}{N}=\Meixner{n}@{x}{-N}{(p-1)^{-1}p} }

Koornwinder Addendum: Meixner

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