GammaRegularized[a, z] == 1 - (z^a/(E^z Gamma[a])) (1/(a - (a z)/(a + z + 1 - ((a + 1) z)/(a + z + 2 - ((a + 2) z)/(a + z + 3 - ((a + 3) z)/(a + z + 4 - ((a + 4) z)/(a + z + 5 + \[Ellipsis])))))))

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 3 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 4 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 5 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> GammaRegularized </ci> <ci> a </ci> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 5 </cn> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29