EllipticTheta[4, z, q] == Sum[(Derivative[0, 0, k][EllipticTheta][4, z, Subscript[q, 0]]/k!) (q - Subscript[q, 0])^k, {k, 0, Infinity}]

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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> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msubsup> <mi> &#977; </mi> <mn> 4 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msubsup> <mi> &#977; </mi> <mn> 4 </mn> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <msub> <mi> q </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

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2007-05-02