Dual Hahn

<< Hahn

Dual Hahn

Meixner-Pollaczek >>

Hypergeometric representation

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Orthogonality relation(s)

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Recurrence relation

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

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

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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Monic recurrence relation

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

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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}=\frac{1}{\pochhammer{\gamma+1}{n}\pochhammer{-N}{n}}\monicdualHahn{n}@@{\lambda(x)}{\gamma}{\delta}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -ny(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)=\frac{x(x+\gamma+\delta+N+1)(x+\delta)}{(2x+\gamma+\delta)(2x+\gamma+\delta+1)}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=\frac{(x+\gamma+1)(x+\gamma+\delta+1)(N-x)}{(2x+\gamma+\delta+1)(2x+\gamma+\delta+2)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \dualHahn{n}@{\lambda(x+1)}{\gamma}{\delta}{N}-\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N} {}=-\frac{n(2x+\gamma+\delta+2)}{(\gamma+1)N}\dualHahn{n-1}@{\lambda(x)}{\gamma+1}{\delta}{N-1} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (x+\gamma)(x+\gamma+\delta)(N+1-x)\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N} {}-x(x+\gamma+\delta+N+1)(x+\delta)\dualHahn{n}@{\lambda(x-1)}{\gamma}{\delta}{N} {}=\gamma(N+1)(2x+\gamma+\delta)\dualHahn{n+1}@{\lambda(x)}{\gamma-1}{\delta}{N+1} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;\gamma,\delta,N)=\frac{(-1)^x\pochhammer{\gamma+1}{x}\pochhammer{\gamma+\delta+1}{x}\pochhammer{-N}{x}} {\pochhammer{\gamma+\delta+N+2}{x}\pochhammer{\delta+1}{x}x!} }}

Rodrigues-type formula

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \nabla_{\lambda}:=\frac{\nabla}{\nabla\lambda(x)} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{N-x}\,\HyperpFq{2}{1}@@{-x,-x-\delta}{\gamma+1}{t} =\sum_{n=0}^N\frac{\pochhammer{-N}{n}}{n!}\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^x\,\HyperpFq{2}{1}@@{x-N,x+\gamma+1}{-\delta-N}{t} =\sum_{n=0}^N\frac{\pochhammer{\gamma+1}{n}\pochhammer{-N}{n}}{\pochhammer{-\delta-N}{n}n!}\dualHahn{n}@{\lambda(x)}{\gamma}{\delta}{N}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \epsilon}} arbitrary

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Limit relations

Racah polynomial to Dual Hahn polynomial

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Dual 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} \dualHahn{n}@{\lambda(x)}{\beta-1}{N(1-c)c^{-1}}{N}=\Meixner{n}@{x}{\beta}{c} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Dual 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}\dualHahn{n}@{\lambda(x)}{pt}{(1-p)t}{N}=\Krawtchouk{n}@{x}{p}{N} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Remark

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

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

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

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\}}}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda(x)=x(x+\gamma+\delta+1)}}

Koornwinder Addendum: Dual Hahn

<< Hahn

Dual Hahn

Meixner-Pollaczek >>