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 JacobiZeta

 http://functions.wolfram.com/08.07.06.0033.01

 Input Form

 JacobiZeta[z, m] == Sum[((1/k!) Derivative[0, k][EllipticE][z, Subscript[m, 0]] - (1/Subscript[c, 0]) Sum[(j + 1) Subscript[d, k - j] Sum[(((-1)^r Binomial[j, r])/(1 + r)) Subscript[p, r, j], {r, 0, j}], {j, 0, k}]) (m - Subscript[m, 0])^k, {k, 0, Infinity}] /; Subscript[a, k] == ((Pochhammer[-(1/2), k] Pochhammer[1/2, k])/k!^2) Hypergeometric2F1[-(1/2) + k, 1/2 + k, 1 + k, Subscript[m, 0]] && Subscript[c, k] == (Pochhammer[1/2, k]^2/k!^2) Hypergeometric2F1[1/2 + k, 1/2 + k, 1 + k, Subscript[m, 0]] && Subscript[b, k] == (1/k!) Derivative[0, k][EllipticF][z, Subscript[m, 0]] && Subscript[d, k] == Sum[Subscript[a, n] Subscript[b, k - n], {n, 0, k}] && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/(Subscript[c, 0] 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["JacobiZeta", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["0", ",", "k"]], "]"]], "[", "EllipticE", "]"]], "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]]]], "-", RowBox[List[FractionBox["1", SubscriptBox["c", "0"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], SubscriptBox["d", RowBox[List["k", "-", "j"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "j"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "r"]], "]"]]]], RowBox[List["1", "+", "r"]]], SubscriptBox["p", RowBox[List["r", ",", "j"]]]]]]]]]]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["m", "0"]]], ")"]], "k"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", SubscriptBox["m", "0"]]], "]"]]]]]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", SubscriptBox["m", "0"]]], "]"]]]]]], " ", "\[And]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["k", "!"]]], RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["0", ",", "k"]], "]"]], "[", "EllipticF", "]"]], "[", RowBox[List["z", ",", SubscriptBox["m", "0"]]], "]"]]]]]], "\[And]", RowBox[List[SubscriptBox["d", "k"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "n"], " ", SubscriptBox["b", RowBox[List["k", "-", "n"]]]]]]]]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SubscriptBox["c", "0"], " ", "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

 Ζ ( z m ) k = 0 ( 1 k ! Ζ ( 0 , k ) ( z m 0 ) - 1 c 0 j = 0 k ( j + 1 ) d k - j r = 0 j ( - 1 ) r ( j r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["r", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] r + 1 p r , j ) ( m - m 0 ) k /; a k ( - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( k ! ) 2 2 F 1 ( k - 1 2 , k + 1 2 ; k + 1 ; m 0 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "-", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SubscriptBox["m", "0"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] c k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 ( k ! ) 2 2 F 1 ( k + 1 2 , k + 1 2 ; k + 1 ; m 0 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SubscriptBox["m", "0"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] b k 1 k ! F ( 0 , k ) ( z m 0 ) d k n = 0 k a n b k - n p j , 0 1 p j , k 1 c 0 k m = 1 k ( j m - k + m ) c m p j , k - m k TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + Ζ ( z m ) k = 0 ( 1 k ! Ζ ( 0 , k ) ( z m 0 ) - 1 c 0 j = 0 k ( j + 1 ) d k - j r = 0 j ( - 1 ) r ( j r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox["r", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] r + 1 p r , j ) ( m - m 0 ) k /; a k ( - 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "k"], Pochhammer] ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ( k ! ) 2 2 F 1 ( k - 1 2 , k + 1 2 ; k + 1 ; m 0 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "-", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SubscriptBox["m", "0"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] c k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] 2 ( k ! ) 2 2 F 1 ( k + 1 2 , k + 1 2 ; k + 1 ; m 0 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]], ",", TagBox[RowBox[List["k", "+", FractionBox["1", "2"]]], Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["k", "+", "1"]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[SubscriptBox["m", "0"], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] b k 1 k ! F ( 0 , k ) ( z m 0 ) d k n = 0 k a n b k - n p j , 0 1 p j , k 1 c 0 k m = 1 k ( j m - k + m ) c m p j , k - m k TagBox["\[DoubleStruckCapitalN]", Function[Integers]] + [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiZeta", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["EllipticE", TagBox[RowBox[List["(", RowBox[List["0", ",", "k"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], RowBox[List["k", "!"]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], " ", SubscriptBox["d", RowBox[List["k", "-", "j"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], "j"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["Binomial", "[", RowBox[List["j", ",", "r"]], "]"]]]], ")"]], " ", SubscriptBox["p", RowBox[List["r", ",", "j"]]]]], RowBox[List["1", "+", "r"]]]]]]]]], SubscriptBox["c", "0"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", SubscriptBox["mm", "0"]]], ")"]], "k"]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", SubscriptBox["mm", "0"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]], "&&", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", SubscriptBox["mm", "0"]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]], "&&", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["EllipticF", TagBox[RowBox[List["(", RowBox[List["0", ",", "k"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["z", ",", SubscriptBox["mm", "0"]]], "]"]], RowBox[List["k", "!"]]]]], "&&", RowBox[List[SubscriptBox["d", "k"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "n"], " ", SubscriptBox["b", RowBox[List["k", "-", "n"]]]]]]]]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", 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"]]]]]]]]], RowBox[List[SubscriptBox["c", "0"], " ", "k"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

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

 2007-05-02