Derivative[n, 0][Hypergeometric0F1Regularized][b, z] == Sum[(1/k!) D[1/Gamma[b + k], {b, n}] z^k, {k, 0, Infinity}] /; Element[n, Integers] && n > 0

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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> <mrow> <msub> <mn> 0 </mn> </msub> <msubsup> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mo> &#8202; </mo> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mi> b </mi> <annotation encoding='Mathematica'> TagBox[TagBox[TagBox[&quot;b&quot;, Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, Hypergeometric0F1, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mfrac> <mn> 1 </mn> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> b </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <mn> 0 </mn> </msub> <msubsup> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mo> &#8202; </mo> <annotation encoding='Mathematica'> TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mi> b </mi> <annotation encoding='Mathematica'> TagBox[TagBox[TagBox[&quot;b&quot;, Hypergeometric0F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox[&quot;z&quot;, Hypergeometric0F1, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mfrac> <mn> 1 </mn> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> b </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>

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2001-10-29