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Transformation formulas - Revision history
2024-03-29T05:22:01Z
Revision history for this page on the wiki
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https://drmf-beta.wmflabs.org/index.php?title=Transformation_formulas&diff=1989&oldid=prev
imported>SeedBot: DRMF
2017-03-05T22:34:29Z
<p>DRMF</p>
<p><b>New page</b></p><div>{{DISPLAYTITLE:Transformation formulas}}<br />
<div id="drmf_head"><br />
<div id="alignleft"> << [[Some integrals|Some integrals]] </div><br />
<div id="aligncenter"> [[Orthogonal_Polynomials#Transformation formulas|Transformation formulas]] </div><br />
<div id="alignright"> [[The q-shifted factorial|The q-shifted factorial]] >> </div><br />
</div><br />
<br />
== Transformation formulas ==<br />
<br />
<math id="KLS:01.07:01">{\displaystyle <br />
\index{Euler's transformation formula}\index{Transformation formula!Euler}<br />
\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{c-a-b}\,\HyperpFq{2}{1}@@{c-a,c-b}{c}{z}<br />
}</math><br /><br />
<math id="KLS:01.07:02">{\displaystyle <br />
\index{Pfaff-Kummer transformation formula}\index{Transformation formula!Pfaff-Kummer}<br />
\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{-a}\,\HyperpFq{2}{1}@@{a,c-b}{c}{\frac{z}{z-1}}<br />
}</math><br /><br />
<math id="KLS:01.07:03">{\displaystyle <br />
\index{Kummer's transformation formula}\index{Transformation formula!Kummer}<br />
\HyperpFq{1}{1}@@{a}{c}{z}=\expe^z\,\HyperpFq{1}{1}@@{c-a}{c}{-z}<br />
}</math><br /><br />
<math id="KLS:01.07:04">{\displaystyle <br />
\HyperpFq{1}{1}@@{-n}{a}{x}=\frac{(-x)^n}{\pochhammer{a}{n}}\,\HyperpFq{2}{0}@@{-n,-a-n+1}{-}{-\frac{1}{x}}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br /><br />
<math id="KLS:01.07:05">{\displaystyle <br />
\HyperpFq{2}{1}@@{-n,b}{c}{x}=\frac{\pochhammer{b}{n}}{\pochhammer{c}{n}}(-x)^n\,\HyperpFq{2}{1}@@{-n,-c-n+1}{-b-n+1}{\frac{1}{x}}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br /><br />
<math id="KLS:01.07:06">{\displaystyle <br />
\index{Whipple's transformation formula}\index{Transformation formula!Whipple}<br />
\HyperpFq{4}{3}@@{-n,a,b,c}{d,e,f}{1}<br />
{}=\frac{\pochhammer{e-a}{n}\pochhammer{f-a}{n}}{\pochhammer{e}{n}\pochhammer{f}{n}}\HyperpFq{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1}<br />
}</math><br />
<div id="drmf_foot"><br />
<div id="alignleft"> << [[Some integrals|Some integrals]] </div><br />
<div id="aligncenter"> [[Orthogonal_Polynomials#Transformation formulas|Transformation formulas]] </div><br />
<div id="alignright"> [[The q-shifted factorial|The q-shifted factorial]] >> </div><br />
</div></div>
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