Continuous dual Hahn
Continuous dual Hahn
Hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}}=\HyperpFq{3}{2}@@{-n,a+\iunit x,a-\iunit x}{a+b,a+c}{1} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_0^{\infty}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2 \ctsdualHahn{m}@{x^2}{a}{b}{c}\ctsdualHahn{n}@{x^2}{a}{b}{c}\,dx {}=\EulerGamma@{n+a+b}\EulerGamma@{n+a+c}\EulerGamma@{n+b+c}n!\,\Kronecker{m}{n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{2\cpi}\int_0^{\infty}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2 \ctsdualHahn{m}@{x^2}{a}{b}{c}\ctsdualHahn{n}@{x^2}{a}{b}{c}\,dx {}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{b-a}\EulerGamma@{c-a}}{\EulerGamma@{-2a}} {}\sum_{\begin{array}{c} {\scriptstyle k=0,1,2\ldots}\\ {\scriptstyle a+k<0}\end{array}} \frac{\pochhammer{2a}{k}\pochhammer{a+1}{k}\pochhammer{a+b}{k}\pochhammer{a+c}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}k!}(-1)^k {} \ctsdualHahn{m}@{-(a+k)^2}{a}{b}{c}\ctsdualHahn{n}@{-(a+k)^2}{a}{b}{c} {}=\EulerGamma@{n+a+b}\EulerGamma@{n+a+c}\EulerGamma@{n+b+c}n!\,\Kronecker{m}{n} }}
Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -\left(a^2+x^2\right)\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}= A_n\normctsdualHahnStilde{n+1}@@{x^2}{a}{b}{c}-\left(A_n+C_n\right)\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}+C_n\normctsdualHahnStilde{n-1}@@{x^2}{a}{b}{c} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=(n+a+b)(n+a+c)}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}:=\normctsdualHahnStilde{n}@{x^2}{a}{b}{c}=\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicctsdualHahn{n}@@{x}{a}{b}{c}=\monicctsdualHahn{n+1}@@{x}{a}{b}{c}+(A_n+C_n-a^2)\monicctsdualHahn{n}@@{x}{a}{b}{c}+A_{n-1}C_n\monicctsdualHahn{n-1}@@{x}{a}{b}{c} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=(n+a+b)(n+a+c)}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsdualHahn{n}@{x^2}{a}{b}{c}=(-1)^n\monicctsdualHahn{n}@@{x^2}{a}{b}{c} }}
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+\iunit)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-\iunit) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B(x)=\frac{(a-\iunit x)(b-\iunit x)(c-\iunit x)}{2\iunit x(2\iunit x-1)}}}
&
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsdualHahn{n}@{(x+\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c}-\ctsdualHahn{n}@{(x-\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c} {}=-2\iunit nx\ctsdualHahn{n-1}@{x^2}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\delta \ctsdualHahn{n}@{x^2}{a}{b}{c}}{\delta x^2}=-n\ctsdualHahn{n-1}@{x^2}{a+\textstyle\frac{1}{2}}{ b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (a-\textstyle\frac{1}{2}-\iunit x)(b-\textstyle\frac{1}{2}-\iunit x) (c-\textstyle\frac{1}{2}-\iunit x) \ctsdualHahn{n}@{(x+\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c} {}-(a-\textstyle\frac{1}{2}+\iunit x)(b-\textstyle\frac{1}{2}+\iunit x) (c-\textstyle\frac{1}{2}+\iunit x) \ctsdualHahn{n}@{(x-\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c} {}=-2\iunit xS_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2} c-\textstyle\frac{1}{2}) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\delta\left[\omega(x;a,b,c)\ctsdualHahn{n}@{x^2}{a}{b}{c}\right]}{\delta x^2} {}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2}) \ctsdualHahn{n+1}@{x^2}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}} }}
Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \omega(x;a,b,c)\ctsdualHahn{n}@{x^2}{a}{b}{c}= \left(\frac{\delta}{\delta x^2}\right)^n\left[\omega(x;a+\textstyle\frac{1}{2}n b+\textstyle\frac{1}{2}n,c+\textstyle\frac{1}{2}n)\right] }}
Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{-c+\iunit x}\,\HyperpFq{2}{1}@@{a+\iunit x,b+\iunit x}{a+b}{t} =\sum_{n=0}^{\infty}\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{-b+\iunit x}\,\HyperpFq{2}{1}@@{a+\iunit x,c+\iunit x}{a+c}{t} =\sum_{n=0}^{\infty}\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+c}{n}n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{-a+\iunit x}\,\HyperpFq{2}{1}@@{b+\iunit x,c+\iunit x}{b+c}{t} =\sum_{n=0}^{\infty}\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{b+c}{n}n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^t\,\HyperpFq{2}{2}@@{a+\iunit x,a-\iunit x}{a+b,a+c}{-t}=\sum_{n=0}^{\infty} \frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{-\gamma}\,\HyperpFq{3}{2}@@{\gamma,a+\iunit x,a-\iunit x}{a+b,a+c}{\frac{t}{t-1}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}n!}t^n }}
Limit relations
Wilson polynomial to Continuous dual Hahn polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{d\rightarrow\infty}\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+d}{n}}=\ctsdualHahn{n}@{x^2}{a}{b}{c} }}
Continuous dual Hahn polynomial to Meixner-Pollaczek polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\frac{\ctsdualHahn{n}@{(x-t)^2}{\lambda+\iunit t}{\lambda-\iunit t}{t\cot@@{\phi}}}{t^nn!} =\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{(\sin@@{\phi})^n} }}
Remark
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\pochhammer{a+b}{n}\pochhammer{a+c}{n}}{\pochhammer{a+b}{k}\pochhammer{a+c}{k}}=\pochhammer{a+b+k}{n-k}\pochhammer{a+c+k}{n-k} }}
Koornwinder Addendum: Continuous dual Hahn
Continuous dual Hahn: Special value
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsdualHahn{n}@{-a^2}{a}{b}{c}=\pochhammer{a+b}{n}\pochhammer{a+c}{n} }}
Uniqueness of orthogonality measure
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{|\ctsdualHahn{n}@{-a^2}{a}{b}{c}|^2}{h_n} = O(n^{2\realpart{a}-1}) \hbox{as $n\to\infty$.} }}