Hahn

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Hahn

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\HyperpFq{3}{2}@@{-n,n+\alpha+\beta+1,-x}{\alpha+1,-N}{1} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n=0,1,2,\ldots,N}}


Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{x=0}^N\binomial{\alpha +x}{x}\binomial{\beta+N-x}{N-x}\Hahn{m}@{x}{\alpha}{\beta}{N}\Hahn{n}@{x}{\alpha}{\beta}{N} {}=\frac{(-1)^n\pochhammer{n+\alpha+\beta+1}{N+1}\pochhammer{\beta+1}{n}n!}{(2n+\alpha+\beta+1)\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}N!}\,\Kronecker{m}{n} }}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -x\Hahn{n}@@{x}{\alpha}{\beta}{N}=A_n\Hahn{n+1}@@{x}{\alpha}{\beta}{N}-\left(A_n+C_n\right)\Hahn{n}@@{x}{\alpha}{\beta}{N}+C_n\Hahn{n-1}@@{x}{\alpha}{\beta}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@@{x}{\alpha}{\beta}{N}:=\Hahn{n}@{x}{\alpha}{\beta}{N} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicHahn{n}@@{x}{\alpha}{\beta}{N}=\monicHahn{n+1}@@{x}{\alpha}{\beta}{N}+\left(A_n+C_n\right)\monicHahn{n}@@{x}{\alpha}{\beta}{N}+A_{n-1}C_n\monicHahn{n-1}@@{x}{\alpha}{\beta}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=\frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(n+\alpha+\beta+1)(n+\alpha+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\frac{\pochhammer{n+\alpha+\beta+1}{n}}{\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}}\monicHahn{n}@@{x}{\alpha}{\beta}{N} }}

Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n(n+\alpha+\beta+1)y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D(x)=x(x-\beta-N-1)}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=(x+\alpha+1)(x-N)}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\Hahn{n}@{x}{\alpha}{\beta}{N}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@{x+1}{\alpha}{\beta}{N}-\Hahn{n}@{x}{\alpha}{\beta}{N} {}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}\Hahn{n-1}@{x}{\alpha+1}{\beta+1}{N-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Delta \Hahn{n}@{x}{\alpha}{\beta}{N}=-\frac{n(n+\alpha+\beta+1)}{(\alpha+1)N}\Hahn{n-1}@{x}{\alpha+1}{\beta+1}{N-1} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (x+\alpha)(N+1-x)\Hahn{n}@{x}{\alpha}{\beta}{N}-x(\beta+N+1-x)\Hahn{n}@{x-1}{\alpha}{\beta}{N} {}=\alpha(N+1)\Hahn{n+1}@{x}{\alpha-1}{\beta-1}{N+1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla\left[\omega(x;\alpha,\beta,N)\Hahn{n}@{x}{\alpha}{\beta}{N}\right] {}=\frac{N+1}{\beta}\omega(x;\alpha-1,\beta-1,N+1)\Hahn{n+1}@{x}{\alpha-1}{\beta-1}{N+1} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;\alpha,\beta,N)=\binomial{\alpha+x}{x}\binomial{\beta+N-x}{N-x}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;\alpha,\beta,N)\Hahn{n}@{x}{\alpha}{\beta}{N} {}=\frac{(-1)^n\pochhammer{\beta+1}{n}}{\pochhammer{-N}{n}}\nabla^n\left[\omega(x;\alpha+n,\beta+n,N-n)\right] }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;\alpha,\beta,N)=\binomial{\alpha+x}{x}\binomial{\beta+N-x}{N-x}}}


Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{1}{1}@@{-x}{\alpha+1}{-t}\,\HyperpFq{1}{1}@@{x-N}{\beta+1}{t} =\sum_{n=0}^N\frac{\pochhammer{-N}{n}}{\pochhammer{\beta+1}{n}n!}\Hahn{n}@{x}{\alpha}{\beta}{N}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{2}{0}@@{-x,-x+\beta+N+1}{-}{-t}\,\HyperpFq{2}{0}@@{x-N,x+\alpha+1}{-}{t} {}=\sum_{n=0}^N\frac{\pochhammer{-N}{n}\pochhammer{\alpha+1}{n}}{n!}\Hahn{n}@{x}{\alpha}{\beta}{N}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left[(1-t)^{-\alpha-\beta-1}\,\HyperpFq{3}{2}@@{\frac{1}{2}(\alpha+\beta+1),\frac{1}{2}(\alpha+\beta+2),-x}{\alpha+1,-N}{-\frac{4t}{(1-t)^2}}\right]_N {}=\sum_{n=0}^N\frac{\pochhammer{\alpha+\beta+1}{n}}{n!}\Hahn{n}@{x}{\alpha}{\beta}{N}t^n }}

Limit relations

Racah polynomial to Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\delta\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{-N-1}{\delta}=\Hahn{n}@{x}{\alpha}{\beta}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\gamma\rightarrow\infty} \Racah{n}@{\lambda(x)}{\alpha}{\beta}{\gamma}{-\beta-N-1}=\Hahn{n}@{x}{\alpha}{\beta}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\delta\rightarrow\infty} \Racah{n}@{\lambda(x)}{-N-1}{\beta+\gamma+N+1}{\gamma}{\delta}=\Hahn{n}@{x}{\gamma}{\beta}{N} }}

Hahn polynomial to Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty} \Hahn{n}@{Nx}{\alpha}{\beta}{N}=\frac{\Jacobi{\alpha}{\beta}{n}@{1-2x}}{\Jacobi{\alpha}{\beta}{n}@{1}} }}

Hahn polynomial to Meixner polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty} \Hahn{n}@{x}{b-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{b}{c} }}

Hahn polynomial to Krawtchouk polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\Hahn{n}@{x}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N} }}

Remark

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\dualHahn{x}@{\lambda(n)}{\alpha}{\beta}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n=0}^N\frac{(2n+\alpha+\beta+1)\pochhammer{\alpha+1}{n}\pochhammer{-N}{n}N!}{(-1)^n\pochhammer{n+\alpha+\beta+1}{N+1}\pochhammer{\beta+1}{n}n!} \Hahn{n}@{x}{\alpha}{\beta}{N}\Hahn{n}@{y}{\alpha}{\beta}{N} {}=\frac{\Kronecker{x}{y}}{\dbinom{\alpha+x}{x}\dbinom{\beta+N-x}{N-x}} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x,y \in \{0,1,2,\ldots,N\}}}


Koornwinder Addendum: Hahn

Hahn: Special values

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@{0}{\alpha}{\beta}{N}=1 \Hahn{n}@{N}{\alpha}{\beta}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@{0}{\alpha}{\beta}{N} =\frac{(-1)^n\pochhammer{\beta+1}{n}}{\pochhammer{\alpha+1}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{2n}@{N}{\alpha}{\alpha}{2N}=\frac{\pochhammer{\frac12}{n}\pochhammer{N+\alpha+1}{n}}{\pochhammer{-N+\frac12}{n}\pochhammer{\alpha+1}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{N}@{x}{\alpha}{\beta}{N}=\frac{\pochhammer{-N-\beta}{x}}{\pochhammer{\alpha+1}{x}}\qquad(x }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{N}@{x}{\alpha}{\beta}{N} =0,1,\ldots,N) }}

Symmetries

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Hahn{n}@{N-x}{\alpha}{\beta}{N}}{\Hahn{n}@{N}{\alpha}{\beta}{N}}=\Hahn{n}@{x}{\beta}{\alpha}{N} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Hahn{N-n}@{x}{\alpha}{\beta}{N}}{\Hahn{N}@{x}{\alpha}{\beta}{N}} =\Hahn{n}@{x}{-N-\beta-1}{-N-\alpha-1}{N} \qquad(x }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Hahn{N-n}@{x}{\alpha}{\beta}{N}}{\Hahn{N}@{x}{\alpha}{\beta}{N}} =0,1,\ldots,N) }}

Duality

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Hahn{n}@{x}{\alpha}{\beta}{N}=\dualHahn{x}@{n(n+\alpha+\beta+1)}{\alpha}{\beta}{N} (n,x\in\{0,1,\ldots N\}) }}

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