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 EllipticThetaPrime

 http://functions.wolfram.com/09.06.06.0014.01

 Input Form

 EllipticThetaPrime[2, z, q] == (((6 Sqrt[Pi] I)/(q - 1)^(3/2)) Sum[Binomial[k + 3/2, k] Sum[(((-1)^j Binomial[k, j])/(2 j + 3)) Subscript[p, j, k] (q - 1)^k E^(z^2/Log[q]) (z + 2 Sum[(-1)^m E^((m^2 Pi^2)/Log[q]) (z Cosh[(2 m Pi z)/Log[q]] + Pi m Sinh[(2 m Pi z)/Log[q]]), {m, 1, Infinity}]), {j, 0, k}], {k, 0, Infinity}])/E^(3 I Pi Floor[3/4 - Arg[q - 1]/(2 Pi)]) /; (Abs[q] < 1 && Abs[q - 1] < 1) && Subscript[c, k] == (-1)^(k - 1)/(k + 1) && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (-(1/k)) Sum[(j m - k + m) Subscript[c, m] Subscript[p, j, k - m], {m, 1, k}] && Element[k, Integers] && k > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticThetaPrime", "[", RowBox[List["2", ",", "z", ",", "q"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["6", " ", SqrtBox["\[Pi]"], "\[ImaginaryI]", " "]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], RowBox[List["3", "/", "2"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "3"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["q", "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", FractionBox["3", "2"]]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], RowBox[List[RowBox[List["2", "j"]], "+", "3"]]], SubscriptBox["p", RowBox[List["j", ",", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], "k"], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["Log", "[", "q", "]"]]]], RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["m", "2"], SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", "q", "]"]]]], RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", "m", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "m", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", "m", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["q", "-", "1"]], "]"]], "<", "1"]]]], ")"]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List["k", "+", "1"]]]]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "k"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], SubscriptBox["c", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]

 MathML Form

 ϑ TagBox["\[CurlyTheta]", EllipticThetaPrime] 2 ( z , q ) 6 π ( q - 1 ) 3 / 2 - 3 π 3 4 - arg ( q - 1 ) 2 π k = 0 ( k + 3 2 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["k", "+", FractionBox["3", "2"]]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] j = 0 k ( - 1 ) j 2 j + 3 ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p j , k ( q - 1 ) k z 2 log ( q ) ( z + 2 m = 1 ( - 1 ) m m 2 π 2 log ( q ) ( z cosh ( 2 m π z log ( q ) ) + m π sinh ( 2 m π z log ( q ) ) ) ) /; ( "\[LeftBracketingBar]" q "\[RightBracketingBar]" < 1 "\[LeftBracketingBar]" q - 1 "\[RightBracketingBar]" < 1 ) c k ( - 1 ) k - 1 k + 1 p j , 0 1 p j , k - 1 k m = 1 k ( j m - k + m ) c m p j , k - m k TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition EllipticThetaPrime 2 z q 6 1 2 q -1 3 2 -1 -3 3 4 -1 q -1 2 -1 k 0 Binomial k 3 2 k j 0 k -1 j 2 j 3 -1 Binomial k j Subscript p j k q -1 k z 2 q -1 z 2 m 1 -1 m m 2 2 q -1 z 2 m z q -1 m 2 m z q -1 q 1 q -1 1 Subscript c k -1 k -1 k 1 -1 Subscript p j 0 1 Subscript p j k -1 1 k -1 m 1 k j m -1 k m Subscript c m Subscript p j k -1 m k SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticThetaPrime", "[", RowBox[List["2", ",", "z_", ",", "q_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["6", " ", SqrtBox["\[Pi]"], " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "3"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["q", "-", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["k", "+", FractionBox["3", "2"]]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["j", ",", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List["z", "+", RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["m", "2"], " ", SuperscriptBox["\[Pi]", "2"]]], RowBox[List["Log", "[", "q", "]"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["z", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["2", " ", "m", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", "m", " ", RowBox[List["Sinh", "[", FractionBox[RowBox[List["2", " ", "m", " ", "\[Pi]", " ", "z"]], RowBox[List["Log", "[", "q", "]"]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "3"]]]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["q", "-", "1"]], ")"]], RowBox[List["3", "/", "2"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "q", "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["q", "-", "1"]], "]"]], "<", "1"]]]], ")"]], "&&", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List["k", "+", "1"]]]]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", SubscriptBox["c", "m"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]], "k"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02