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=== Section 9.1 [[Wilson|Wilson]] ===
=== Section 9.1 [[Wilson|Wilson]] ===
==== [[Definition:normWilsonWtilde|normWilsonWtilde]] ====
Occurences:
[https://drmf-beta.wmflabs.org/w/index.php?oldid=602#math.602.3 <math display=block>{{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=:(-1)^{n}\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@{x^{2}}{a}{b}{c}{d}.}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=607#math.607.3 <math display=block>{{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}:=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=625#math.625.0 <math display=block>{{{\Wilson{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=625#math.625.1 <math display=block>{{{\Wilson{n}@{x}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=625#math.625.2 <math display=block>{{{\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{}:=\HyperpFq{4}{3}@{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=648#KLS:14.01:37 <math display=block>{{{\lim_{q\rightarrow 1}\frac{\AskeyWilson{n}@{\frac{1}{2}\left(q^{ix}+q^{-ix}\right)}{q^{a}}{q^{b}}{q^{c}}{q^{d}}{q}}{(1-q)^{3n}}=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=703#math.703.0 <math display=block>{{{\lim_{q\rightarrow 1}\frac{\AskeyWilson{n}@{\frac{1}{2}\left(q^{ix}+q^{-ix}\right)}{q^{a}}{q^{b}}{q^{c}}{q^{d}}{q}}{(1-q)^{3n}}=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:01 <math display=block>{{{\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{}=\HyperpFq{4}{3}@{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:02 <math display=block>{{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:04 <math display=block>{{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b-a}\EulerGamma@{c-a}\EulerGamma@{d-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+d}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}\pochhammer{a-d+1}{k}k!}{}\Wilson{m}@{-(a+k)^{2}}{a}{b}{c}{d}\Wilson{n}@{-(a+k)^{2}}{a}{b}{c}{d}{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:06 <math display=block>{{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}:=\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:10 <math display=block>{{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=(-1)^{n}\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@{x^{2}}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:15 <math display=block>{{{\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}-\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2inx(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:16 <math display=block>{{{\frac{\delta\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\delta x^{2}}{}=-n(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:17 <math display=block>{{{(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix)(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix)\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix)(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix)\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2ix\Wilson{n+1}@{x^{2}}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:18 <math display=block>{{{\frac{\delta\left[\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}\right]}{\delta x^{2}}{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\Wilson{n+1}@{x^{2}}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:20 <math display=block>{{{\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}{}=\left(\frac{\delta}{\delta x^{2}}\right)^{n}\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}nc+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:21 <math display=block>{{{\HyperpFq{2}{1}@{a+ix,b+ix}{a+b}{t}\,\HyperpFq{2}{1}@{c-ix,d-ix}{c+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+b}{n}\pochhammer{c+d}{n}n!}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:22 <math display=block>{{{\HyperpFq{2}{1}@{a+ix,c+ix}{a+c}{t}\,\HyperpFq{2}{1}@{b-ix,d-ix}{b+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+c}{n}\pochhammer{b+d}{n}n!}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:23 <math display=block>{{{\HyperpFq{2}{1}@{a+ix,d+ix}{a+d}{t}\,\HyperpFq{2}{1}@{b-ix,c-ix}{b+c}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+d}{n}\pochhammer{b+c}{n}n!}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:24 <math display=block>{{{(1-t)^{1-a-b-c-d}{}\HyperpFq{4}{3}@{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix}{a+b,a+c,a+d}{-\frac{4t}{(1-t)^{2}}}{}=\sum_{n=0}^{\infty}\frac{\pochhammer{a+b+c+d-1}{n}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}n!}\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:25 <math display=block>{{{\lim_{d\rightarrow\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+d}{n}}=\ctsdualHahn{n}@{x^{2}}{a}{b}{c}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:26 <math display=block>{{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:27 <math display=block>{{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^{2}}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+it}{\frac{1}{2}(\beta+1}-it)}{t^{2n}n!}{}=\Jacobi{\alpha}{\beta}{n}@{x}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:34 <math display=block>{{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:36 <math display=block>{{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=\frac{\pochhammer{a-ix}{n}\pochhammer{b-ix}{n}\pochhammer{c-ix}{n}\pochhammer{d-ix}{n}}{\pochhammer{-2ix}{n}}\HyperpFq{7}{6}@{2ix-n,ix-\frac{1}{2}n+1,a+ix,b+ix,c+ix,d+ix,-n}{ix-\frac{1}{2}n,1-n-a+ix,1-n-b+ix,1-n-c+ix,1-n-d+ix}{1}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:37 <math display=block>{{{\Wilson{n}@{-a^{2}}{a}{b}{c}{d}=\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#KLS:09.01:38 <math display=block>{{{\frac{|\Wilson{n}@{-a^{2}}{a}{b}{c}{d}|^{2}}{h_{n}}=O(n^{4\realpart{a}-1})\hbox{as $n\to\infty$.}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1465#math.1465.15 <math display=block>{{{ y(x)=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1466#KLS:09.02:37 <math display=block>{{{\Racah{n}@{\lambda(-a+ix)}{a+b-1}{c+d-1}{a+d-1}{a-d}{}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1466#KLS:09.02:38 <math display=block>{{{\Racah{n}@{x(x-N+\delta)}{\alpha}{\beta}{-N-1}{\delta}=\frac{\Wilson{n}@{-(x+\frac{1}{2}(\delta-N))^{2}}{\frac{1}{2}(\delta-N)}{\alpha+1-\frac{1}{2}(\delta-N)}{\beta+\frac{1}{2}(\delta+N)+1}{-\frac{1}{2}(\delta+N)}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{-N}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1467#KLS:09.03:25 <math display=block>{{{\lim_{d\rightarrow\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+d}{n}}=\ctsdualHahn{n}@{x^{2}}{a}{b}{c}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1468#KLS:09.04:22 <math display=block>{{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1472#KLS:09.08:17 <math display=block>{{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^{2}}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+it}{\frac{1}{2}(\beta+1}-it)}{t^{2n}n!}=\Jacobi{\alpha}{\beta}{n}@{x}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1481#math.1481.0 <math display=block>{{{\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{}=\HyperpFq{4}{3}@{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1482#math.1482.0 <math display=block>{{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1484#math.1484.0 <math display=block>{{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b-a}\EulerGamma@{c-a}\EulerGamma@{d-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+d}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}\pochhammer{a-d+1}{k}k!}{}\Wilson{m}@{-(a+k)^{2}}{a}{b}{c}{d}\Wilson{n}@{-(a+k)^{2}}{a}{b}{c}{d}{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1486#math.1486.0 <math display=block>{{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}:=\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1488#math.1488.0 <math display=block>{{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=(-1)^{n}\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@{x^{2}}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1489#math.1489.3 <math display=block>{{{ y(x)=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1490#math.1490.0 <math display=block>{{{\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}-\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2inx(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1491#math.1491.0 <math display=block>{{{\frac{\delta\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\delta x^{2}}{}=-n(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1492#math.1492.0 <math display=block>{{{(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix)(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix)\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix)(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix)\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2ix\Wilson{n+1}@{x^{2}}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1493#math.1493.0 <math display=block>{{{\frac{\delta\left[\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}\right]}{\delta x^{2}}{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\Wilson{n+1}@{x^{2}}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1494#math.1494.0 <math display=block>{{{\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}{}=\left(\frac{\delta}{\delta x^{2}}\right)^{n}\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}nc+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1495#math.1495.0 <math display=block>{{{\HyperpFq{2}{1}@{a+ix,b+ix}{a+b}{t}\,\HyperpFq{2}{1}@{c-ix,d-ix}{c+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+b}{n}\pochhammer{c+d}{n}n!}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1496#math.1496.0 <math display=block>{{{\HyperpFq{2}{1}@{a+ix,c+ix}{a+c}{t}\,\HyperpFq{2}{1}@{b-ix,d-ix}{b+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+c}{n}\pochhammer{b+d}{n}n!}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1497#math.1497.0 <math display=block>{{{\HyperpFq{2}{1}@{a+ix,d+ix}{a+d}{t}\,\HyperpFq{2}{1}@{b-ix,c-ix}{b+c}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+d}{n}\pochhammer{b+c}{n}n!}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1498#math.1498.0 <math display=block>{{{(1-t)^{1-a-b-c-d}{}\HyperpFq{4}{3}@{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix}{a+b,a+c,a+d}{-\frac{4t}{(1-t)^{2}}}{}=\sum_{n=0}^{\infty}\frac{\pochhammer{a+b+c+d-1}{n}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}n!}\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1499#math.1499.0 <math display=block>{{{\lim_{d\rightarrow\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+d}{n}}=\ctsdualHahn{n}@{x^{2}}{a}{b}{c}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1500#math.1500.0 <math display=block>{{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1501#math.1501.0 <math display=block>{{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^{2}}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+it}{\frac{1}{2}(\beta+1}-it)}{t^{2n}n!}{}=\Jacobi{\alpha}{\beta}{n}@{x}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1508#math.1508.0 <math display=block>{{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1510#math.1510.0 <math display=block>{{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=\frac{\pochhammer{a-ix}{n}\pochhammer{b-ix}{n}\pochhammer{c-ix}{n}\pochhammer{d-ix}{n}}{\pochhammer{-2ix}{n}}\HyperpFq{7}{6}@{2ix-n,ix-\frac{1}{2}n+1,a+ix,b+ix,c+ix,d+ix,-n}{ix-\frac{1}{2}n,1-n-a+ix,1-n-b+ix,1-n-c+ix,1-n-d+ix}{1}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1511#math.1511.0 <math display=block>{{{\Wilson{n}@{-a^{2}}{a}{b}{c}{d}=\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1512#math.1512.0 <math display=block>{{{\frac{|\Wilson{n}@{-a^{2}}{a}{b}{c}{d}|^{2}}{h_{n}}=O(n^{4\realpart{a}-1})\hbox{as $n\to\infty$.}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1540#math.1540.0 <math display=block>{{{\Racah{n}@{\lambda(-a+ix)}{a+b-1}{c+d-1}{a+d-1}{a-d}{}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1541#math.1541.0 <math display=block>{{{\Racah{n}@{x(x-N+\delta)}{\alpha}{\beta}{-N-1}{\delta}=\frac{\Wilson{n}@{-(x+\frac{1}{2}(\delta-N))^{2}}{\frac{1}{2}(\delta-N)}{\alpha+1-\frac{1}{2}(\delta-N)}{\beta+\frac{1}{2}(\delta+N)+1}{-\frac{1}{2}(\delta+N)}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{-N}{n}}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1560#math.1560.0 <math display=block>{{{\lim_{d\rightarrow\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+d}{n}}=\ctsdualHahn{n}@{x^{2}}{a}{b}{c}}}}</math>]
[https://drmf-beta.wmflabs.org/w/index.php?oldid=1580#math.1580.0 <math display=block>{{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}</math>]
=== Section 9.2 [[Racah|Racah]] ===


=== Section 9.1 [[Racah|Racah]]
=== Section 9.3 [[Continuous dual Hahn|Continuous dual Hahn]] ===


=== Section 9.1 [[Continuous dual Hahn|Continuous dual Hahn]]
=== Section 9.4 [[Continuous Hahn|Continuous Hahn]] ===


=== Section 9.1 [[Continuous Hahn|Continuous Hahn]] ===
=== Section 9.5 [[Hahn|Hahn]] ===


=== Section 9.2 [[Hahn|Hahn]] ===
=== Section 9.6 [[Dual Hahn|Dual Hahn]] ===


=== Section 9.3 [[Dual Hahn|Dual Hahn]] ===
=== Section 9.7 [[Meixner-Pollaczek|Meixner-Pollaczek]] ===


=== Section 9.4 [[Meixner-Pollaczek|Meixner-Pollaczek]] ===
=== Section 9.8 [[Jacobi|Jacobi]] ===


=== Section 9.5 [[Jacobi|Jacobi]] ===
=== Section 9.9[[Jacobi: Special cases|Jacobi: Special cases]] ===


=== Section 9.6[[Jacobi: Special cases|Jacobi: Special cases]] ===
=== Section 9.10 [[Pseudo Jacobi|Pseudo Jacobi]] ===


=== Section 9.7 [[Pseudo Jacobi|Pseudo Jacobi]] ===
=== Section 9.11 [[Meixner|Meixner]] ===


=== Section 9.8 [[Meixner|Meixner]] ===
=== Section 9.12 [[Krawtchouk|Krawtchouk]] ===


=== Section 9.9 [[Krawtchouk|Krawtchouk]] ===
=== Section 9.13 [[Laguerre|Laguerre]] ===


=== Section 9.10 [[Laguerre|Laguerre]] ===
=== Section 9.14 [[Bessel|Bessel]] ===


=== Section 9.11 [[Bessel|Bessel]] ===
=== Section 9.15 [[Charlier|Charlier]] ===


=== Section 9.12 [[Charlier|Charlier]] ===
=== Section 9.16 [[Hermite|Hermite]] ===
 
=== Section 9.13 [[Hermite|Hermite]] ===

Revision as of 18:11, 13 July 2017

Symbols in KLS Chapter 9

Section 9.1 Wilson

normWilsonWtilde

Occurences: Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=:(-1)^{n}\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@{x^{2}}{a}{b}{c}{d}.}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}:=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{x}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{}:=\HyperpFq{4}{3}@{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{q\rightarrow 1}\frac{\AskeyWilson{n}@{\frac{1}{2}\left(q^{ix}+q^{-ix}\right)}{q^{a}}{q^{b}}{q^{c}}{q^{d}}{q}}{(1-q)^{3n}}=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{q\rightarrow 1}\frac{\AskeyWilson{n}@{\frac{1}{2}\left(q^{ix}+q^{-ix}\right)}{q^{a}}{q^{b}}{q^{c}}{q^{d}}{q}}{(1-q)^{3n}}=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{}=\HyperpFq{4}{3}@{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b-a}\EulerGamma@{c-a}\EulerGamma@{d-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+d}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}\pochhammer{a-d+1}{k}k!}{}\Wilson{m}@{-(a+k)^{2}}{a}{b}{c}{d}\Wilson{n}@{-(a+k)^{2}}{a}{b}{c}{d}{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}:=\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=(-1)^{n}\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@{x^{2}}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}-\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2inx(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{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 {{{\frac{\delta\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\delta x^{2}}{}=-n(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{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 {{{(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix)(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix)\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix)(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix)\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2ix\Wilson{n+1}@{x^{2}}{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 {{{\frac{\delta\left[\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}\right]}{\delta x^{2}}{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\Wilson{n+1}@{x^{2}}{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 {{{\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}{}=\left(\frac{\delta}{\delta x^{2}}\right)^{n}\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}nc+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\HyperpFq{2}{1}@{a+ix,b+ix}{a+b}{t}\,\HyperpFq{2}{1}@{c-ix,d-ix}{c+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+b}{n}\pochhammer{c+d}{n}n!}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\HyperpFq{2}{1}@{a+ix,c+ix}{a+c}{t}\,\HyperpFq{2}{1}@{b-ix,d-ix}{b+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+c}{n}\pochhammer{b+d}{n}n!}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\HyperpFq{2}{1}@{a+ix,d+ix}{a+d}{t}\,\HyperpFq{2}{1}@{b-ix,c-ix}{b+c}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+d}{n}\pochhammer{b+c}{n}n!}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{(1-t)^{1-a-b-c-d}{}\HyperpFq{4}{3}@{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix}{a+b,a+c,a+d}{-\frac{4t}{(1-t)^{2}}}{}=\sum_{n=0}^{\infty}\frac{\pochhammer{a+b+c+d-1}{n}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}n!}\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^{2}}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+it}{\frac{1}{2}(\beta+1}-it)}{t^{2n}n!}{}=\Jacobi{\alpha}{\beta}{n}@{x}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=\frac{\pochhammer{a-ix}{n}\pochhammer{b-ix}{n}\pochhammer{c-ix}{n}\pochhammer{d-ix}{n}}{\pochhammer{-2ix}{n}}\HyperpFq{7}{6}@{2ix-n,ix-\frac{1}{2}n+1,a+ix,b+ix,c+ix,d+ix,-n}{ix-\frac{1}{2}n,1-n-a+ix,1-n-b+ix,1-n-c+ix,1-n-d+ix}{1}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{-a^{2}}{a}{b}{c}{d}=\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{|\Wilson{n}@{-a^{2}}{a}{b}{c}{d}|^{2}}{h_{n}}=O(n^{4\realpart{a}-1})\hbox{as $n\to\infty$.}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{ y(x)=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Racah{n}@{\lambda(-a+ix)}{a+b-1}{c+d-1}{a+d-1}{a-d}{}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Racah{n}@{x(x-N+\delta)}{\alpha}{\beta}{-N-1}{\delta}=\frac{\Wilson{n}@{-(x+\frac{1}{2}(\delta-N))^{2}}{\frac{1}{2}(\delta-N)}{\alpha+1-\frac{1}{2}(\delta-N)}{\beta+\frac{1}{2}(\delta+N)+1}{-\frac{1}{2}(\delta+N)}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{-N}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^{2}}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+it}{\frac{1}{2}(\beta+1}-it)}{t^{2n}n!}=\Jacobi{\alpha}{\beta}{n}@{x}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{}=\HyperpFq{4}{3}@{-n,n+a+b+c+d-1,a+ix,a-ix}{a+b,a+c,a+d}{1}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\EulerGamma@{a+ix}\EulerGamma@{b+ix}\EulerGamma@{c+ix}\EulerGamma@{d+ix}}{\EulerGamma@{2ix}}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b-a}\EulerGamma@{c-a}\EulerGamma@{d-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+d}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}\pochhammer{a-d+1}{k}k!}{}\Wilson{m}@{-(a+k)^{2}}{a}{b}{c}{d}\Wilson{n}@{-(a+k)^{2}}{a}{b}{c}{d}{}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}:=\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=(-1)^{n}\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@{x^{2}}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{ y(x)=\Wilson{n}@{x^{2}}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}-\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2inx(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{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 {{{\frac{\delta\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\delta x^{2}}{}=-n(n+a+b+c+d-1)\Wilson{n-1}@{x^{2}}{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 {{{(a-\textstyle\frac{1}{2}-ix)(b-\textstyle\frac{1}{2}-ix)(c-\textstyle\frac{1}{2}-ix)(d-\textstyle\frac{1}{2}-ix)\Wilson{n}@{(x+\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}-(a-\textstyle\frac{1}{2}+ix)(b-\textstyle\frac{1}{2}+ix)(c-\textstyle\frac{1}{2}+ix)(d-\textstyle\frac{1}{2}+ix)\Wilson{n}@{(x-\textstyle\frac{1}{2}i)^{2}}{a}{b}{c}{d}{}=-2ix\Wilson{n+1}@{x^{2}}{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 {{{\frac{\delta\left[\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}\right]}{\delta x^{2}}{}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2})\Wilson{n+1}@{x^{2}}{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 {{{\omega(x;a,b,c,d)\Wilson{n}@{x^{2}}{a}{b}{c}{d}{}=\left(\frac{\delta}{\delta x^{2}}\right)^{n}\left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}nc+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right]}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\HyperpFq{2}{1}@{a+ix,b+ix}{a+b}{t}\,\HyperpFq{2}{1}@{c-ix,d-ix}{c+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+b}{n}\pochhammer{c+d}{n}n!}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\HyperpFq{2}{1}@{a+ix,c+ix}{a+c}{t}\,\HyperpFq{2}{1}@{b-ix,d-ix}{b+d}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+c}{n}\pochhammer{b+d}{n}n!}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\HyperpFq{2}{1}@{a+ix,d+ix}{a+d}{t}\,\HyperpFq{2}{1}@{b-ix,c-ix}{b+c}{t}=\sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}{\pochhammer{a+d}{n}\pochhammer{b+c}{n}n!}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{(1-t)^{1-a-b-c-d}{}\HyperpFq{4}{3}@{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+ix,a-ix}{a+b,a+c,a+d}{-\frac{4t}{(1-t)^{2}}}{}=\sum_{n=0}^{\infty}\frac{\pochhammer{a+b+c+d-1}{n}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}n!}\Wilson{n}@{x^{2}}{a}{b}{c}{d}t^{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^{2}}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+it}{\frac{1}{2}(\beta+1}-it)}{t^{2n}n!}{}=\Jacobi{\alpha}{\beta}{n}@{x}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\normWilsonWtilde{n}@{x^{2}}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{x^{2}}{a}{b}{c}{d}=\frac{\pochhammer{a-ix}{n}\pochhammer{b-ix}{n}\pochhammer{c-ix}{n}\pochhammer{d-ix}{n}}{\pochhammer{-2ix}{n}}\HyperpFq{7}{6}@{2ix-n,ix-\frac{1}{2}n+1,a+ix,b+ix,c+ix,d+ix,-n}{ix-\frac{1}{2}n,1-n-a+ix,1-n-b+ix,1-n-c+ix,1-n-d+ix}{1}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Wilson{n}@{-a^{2}}{a}{b}{c}{d}=\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\frac{|\Wilson{n}@{-a^{2}}{a}{b}{c}{d}|^{2}}{h_{n}}=O(n^{4\realpart{a}-1})\hbox{as $n\to\infty$.}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Racah{n}@{\lambda(-a+ix)}{a+b-1}{c+d-1}{a+d-1}{a-d}{}=\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\Racah{n}@{x(x-N+\delta)}{\alpha}{\beta}{-N-1}{\delta}=\frac{\Wilson{n}@{-(x+\frac{1}{2}(\delta-N))^{2}}{\frac{1}{2}(\delta-N)}{\alpha+1-\frac{1}{2}(\delta-N)}{\beta+\frac{1}{2}(\delta+N)+1}{-\frac{1}{2}(\delta+N)}}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+\delta+1}{n}\pochhammer{-N}{n}}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}}}}} Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {{{\lim_{t\rightarrow\infty}\frac{\Wilson{n}@{(x+t)^{2}}{a-it}{b-it}{c+it}{d+it}}{(-2t)^{n}n!}=\ctsHahn{n}@{x}{a}{b}{c}{d}}}}}

Section 9.2 Racah

Section 9.3 Continuous dual Hahn

Section 9.4 Continuous Hahn

Section 9.5 Hahn

Section 9.6 Dual Hahn

Section 9.7 Meixner-Pollaczek

Section 9.8 Jacobi

Section 9.9Jacobi: Special cases

Section 9.10 Pseudo Jacobi

Section 9.11 Meixner

Section 9.12 Krawtchouk

Section 9.13 Laguerre

Section 9.14 Bessel

Section 9.15 Charlier

Section 9.16 Hermite