D[InverseGammaRegularized[a, Subscript[z, 1], Subscript[z, 2]], {Subscript[z, 2], n}] == KroneckerDelta[n] w + ((Gamma[a] E^w)/w^(a - 1))^n Sum[\[Ellipsis] Sum[KroneckerDelta[Sum[(i - 1) Subscript[j, i], {i, 2, n}], n - 1] (-1)^Sum[Subscript[j, i], {i, 2, n}] (n - 1 + Sum[Subscript[j, i], {i, 2, n}])! Product[(1/Subscript[j, i]!) ((Gamma[a + 1]/i!) E^w w^(1 - a))^ Subscript[j, i] (GammaRegularized[a, Subscript[z, 1]] KroneckerDelta[i] + (a Sum[(-1)^(i - k) Binomial[i, k] Pochhammer[1 - a - k, i - 1] GammaRegularized[a + k, w], {k, 0, i}])/w^i)^Subscript[j, i], {i, 2, n}], {Subscript[j, n], 0, n}], {Subscript[j, 2], 0, n}] /; w == InverseGammaRegularized[a, Subscript[z, 1], Subscript[z, 2]] && Element[n, Integers] && n >= 0

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</mo> <msub> <mi> j </mi> <mi> i </mi> </msub> </mrow> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> j </mi> <mi> i </mi> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> j </mi> <mi> i </mi> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> j </mi> <mi> i </mi> </msub> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mi> w </mi> </msup> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> </msup> </mrow> <mrow> <mi> i </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <msub> <mi> j </mi> <mi> i </mi> </msub> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mo> - </mo> <mi> i </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> i </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> i </mi> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> i </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;i&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> &#8290; 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</ms> <apply> <ci> RowBox </ci> <list> <ms> i </ms> <ms> = </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> i </ms> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> i </ms> <ms> = </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> i </ms> </apply> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> RowBox </ci> <list> <ms> i </ms> <ms> = </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> i </ms> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; 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2007-05-02