Continuous Hahn
Continuous Hahn
Hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsHahn{n}@{x}{a}{b}{c}{d} {}=\iunit^n\frac{\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{n!}\,\HyperpFq{3}{2}@@{-n,n+a+b+c+d-1,a+\iunit x}{a+c,a+d}{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_{-\infty}^{\infty}\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c-\iunit x}\EulerGamma@{d-\iunit x} \ctsHahn{m}@{x}{a}{b}{c}{d}\ctsHahn{n}@{x}{a}{b}{c}{d}\,dx {}=\frac{\EulerGamma@{n+a+c}\EulerGamma@{n+a+d}\EulerGamma@{n+b+c}\EulerGamma@{n+b+d}} {(2n+a+b+c+d-1)\EulerGamma@{n+a+b+c+d-1}n!}\,\Kronecker{m}{n} }}
Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (a+\iunit x)\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}=A_n\normctsHahnptilde{n+1}@@{x}{a}{b}{c}{d}-\left(A_n+C_n\right)\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}+C_n\normctsHahnptilde{n-1}@@{x}{a}{b}{c}{d} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normctsHahnptilde{n}@@{x}{a}{b}{c}{d}:=\normctsHahnptilde{n}@{x}{a}{b}{c}{d}=\frac{n!}{\iunit^n\pochhammer{a+c}{n}\pochhammer{a+d}{n}}\ctsHahn{n}@{x}{a}{b}{c}{d} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicctsHahn{n}@@{x}{a}{b}{c}{d}=\monicctsHahn{n+1}@@{x}{a}{b}{c}{d}+\iunit(A_n+C_n+a)\monicctsHahn{n}@@{x}{a}{b}{c}{d}-A_{n-1}C_n\monicctsHahn{n-1}@@{x}{a}{b}{c}{d} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=-\frac{(n+a+b+c+d-1)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsHahn{n}@{x}{a}{b}{c}{d}=\frac{\pochhammer{n+a+b+c+d-1}{n}}{n!}\monicctsHahn{n}@@{x}{a}{b}{c}{d} }}
Difference equation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n(n+a+b+c+d-1)y(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)=(c-\iunit x)(d-\iunit x)}}
&
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsHahn{n}@{x+\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d}-\ctsHahn{n}@{x-\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d} {}=\iunit(n+a+b+c+d-1)\ctsHahn{n-1}@{x}{a+\textstyle\frac{1}{2}}{ b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\delta \ctsHahn{n}@{x}{a}{b}{c}{d}}{\delta x}=(n+a+b+c+d-1) \ctsHahn{n-1}@{x}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{ c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}} }}
Backward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (c-\textstyle\frac{1}{2}-\iunit x)(d-\textstyle\frac{1}{2}-\iunit x) \ctsHahn{n}@{x+\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d} {}-(a-\textstyle\frac{1}{2}+\iunit x)(b-\textstyle\frac{1}{2}+\iunit x) \ctsHahn{n}@{x-\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d} {}=\frac{n+1}{\iunit}\ctsHahn{n+1}@{x}{a-\textstyle\frac{1}{2}}{ b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\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,d)\ctsHahn{n}@{x}{a}{b}{c}{d}\right]}{\delta x} {}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2} c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) {} \ctsHahn{n+1}@{x}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\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,d)\ctsHahn{n}@{x}{a}{b}{c}{d} {}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n \left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n c+\textstyle\frac{1}{2}n,d+\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 \HyperpFq{1}{1}@@{a+\iunit x}{a+c}{-\iunit t}\,\HyperpFq{1}{1}@@{d-\iunit x}{b+d}{\iunit t}= \sum_{n=0}^{\infty}\frac{\ctsHahn{n}@{x}{a}{b}{c}{d}}{\pochhammer{a+c}{n}\pochhammer{b+d}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{1}{1}@@{a+\iunit x}{a+d}{-\iunit t}\,\HyperpFq{1}{1}@@{c-\iunit x}{b+c}{\iunit t}= \sum_{n=0}^{\infty}\frac{\ctsHahn{n}@{x}{a}{b}{c}{d}}{\pochhammer{a+d}{n}\pochhammer{b+c}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-t)^{1-a-b-c-d}\,\HyperpFq{3}{2}@@{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+\iunit x}{a+c,a+d}{-\frac{4t}{(1-t)^2}} {}=\sum_{n=0}^{\infty} \frac{\pochhammer{a+b+c+d-1}{n}}{\pochhammer{a+c}{n}\pochhammer{a+d}{n}\iunit^n}\ctsHahn{n}@{x}{a}{b}{c}{d}t^n }}
Limit relations
Wilson polynomial to Continuous Hahn polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty} \frac{\Wilson{n}@{(x+t)^2}{a-\iunit t}{b-\iunit t}{c+\iunit t}{d+\iunit t}}{(-2t)^nn!}=\ctsHahn{n}@{x}{a}{b}{c}{d} }}
Continuous 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{\ctsHahn{n}@{x+t}{\lambda-\iunit t}{t\tan@@{\phi}}{\lambda+\iunit t}{t\tan@@{\phi}}}{t^n} =\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{(\cos@@{\phi})^n} }}
Continuous Hahn polynomial to Jacobi polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty} \frac{\ctsHahn{n}@{\frac{1}{2}xt}{\frac{1}{2}(\alpha+1-\iunit t)}{\frac{1}{2}(\beta+1+\iunit t)}{ \frac{1}{2}(\alpha+1+\iunit t)}{\frac{1}{2}(\beta+1-\iunit t})}{t^n} {}=\Jacobi{\alpha}{\beta}{n}@{x} }}
Continuous Hahn polynomial to Pseudo Jacobi polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\frac{\ctsHahn{n}@{xt}{\frac{1}{2}(-N+\iunit\nu-2t)}{\frac{1}{2}(-N-\iunit\nu+2t)}{ \frac{1}{2}(-N+\iunit\nu-2t)}{\frac{1}{2}(-N-\iunit\nu+2t})}{t^n} {}=\frac{\pochhammer{n-2N-1}{n}}{n!}\pseudoJacobi{n}@{x}{\nu}{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 Hahn
Continuous Hahn: Special cases
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsHahn{n}@{x}{a}{a+\frac12}{a}{a+\frac12}= \frac{\pochhammer{2a}{n} \pochhammer{2a+\frac12}{n}}{\pochhammer{4a}{n}} \MeixnerPollaczek{2a}{n}@{2x}{\frac12\cpi} }}