EllipticTheta[4, x, q^a] EllipticTheta[4, y, q^b] == Sum[(-1)^r q^(b r^2) E^(2 r I y) EllipticTheta[3, x + y + r b Pi \[Tau], q^(a + b)] EllipticTheta[3, a y - b x + (Pi/2) Mod[a - b, 2] + r a b Pi \[Tau], q^(a b (a + b))], {r, 0, a + b - 1}] /; Element[{a, b}, Integers] && a > 0 && b > 0 && q == E^(I Pi \[Tau])

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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> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <msup> <mi> q </mi> <mi> a </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mi> y </mi> <mo> , </mo> <msup> <mi> q </mi> <mi> b </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> r </mi> </msup> <mo> &#8290; </mo> <msup> <mi> q </mi> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> r </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> y </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> r </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> </mrow> <mo> , </mo> <msup> <mi> q </mi> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> r </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 2 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <plus /> <ci> $CellContext`a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> $CellContext`b </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> , </mo> <msup> <mi> q </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> } </mo> </mrow> <mo> &#8712; </mo> <msup> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> q </mi> <mo> &#10869; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#964; </mi> </mrow> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <ci> x </ci> <apply> <power /> <ci> q </ci> <ci> a </ci> </apply> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <ci> y </ci> <apply> <power /> <ci> q </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> <imaginaryi /> <ci> y </ci> </apply> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> x </ci> <ci> y </ci> <apply> <times /> <ci> b </ci> <pi /> <ci> r </ci> <ci> &#964; </ci> </apply> </apply> <apply> <power /> <ci> q </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> x </ci> </apply> <apply> <times /> <ci> a </ci> <ci> y </ci> </apply> <apply> <times /> <ci> a </ci> <ci> b </ci> <pi /> <ci> r </ci> <ci> &#964; </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <pi /> <apply> <rem /> <apply> <plus /> <ci> $CellContext`a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> $CellContext`b </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <ci> a </ci> <ci> b </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <list> <ci> a </ci> <ci> b </ci> </list> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <eq /> <ci> q </ci> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#964; </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29