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 EllipticPi

 http://functions.wolfram.com/08.06.20.0011.01

 Input Form

 D[EllipticPi[n, z, m], {z, \[Alpha]}] == Sqrt[Pi] z^(1 - \[Alpha]) 2^(\[Alpha] - 1) Sum[(((-1)^(k + j + l) 2^(-2 k - 2 j - 2 l) m^l n^k)/ ((1 + 2 k + 2 j + 2 l) j! l!)) Pochhammer[1/2, j] Pochhammer[1/2, l] Sum[(-1)^p Binomial[1 + 2 k + 2 j + 2 l, p] (1 + 2 k + 2 j + 2 l - 2 p) HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/2}, -(((1 + 2 k + 2 j + 2 l - 2 p)^2 z^2)/4)], {p, 0, k + j + l}], {j, 0, Infinity}, {l, 0, Infinity}, {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "j", "+", "l"]]], SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "k"]], "-", RowBox[List["2", "j"]], "-", RowBox[List["2", "l"]]]]], SuperscriptBox["m", "l"], SuperscriptBox["n", "k"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", "k"]], "+", RowBox[List["2", "j"]], "+", RowBox[List["2", "l"]]]], ")"]], RowBox[List["j", "!"]], " ", RowBox[List["l", "!"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "l"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["k", "+", "j", "+", "l"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", "k"]], "+", RowBox[List["2", "j"]], "+", RowBox[List["2", "l"]]]], ",", "p"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", "k"]], "+", RowBox[List["2", "j"]], "+", RowBox[List["2", "l"]], "-", RowBox[List["2", " ", "p"]]]], ")"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", "k"]], "+", RowBox[List["2", "j"]], "+", RowBox[List["2", "l"]], "-", RowBox[List["2", " ", "p"]]]], ")"]], "2"], SuperscriptBox["z", "2"]]], "4"]]]]], "]"]]]]]]]]]]]]]]]]]]]]

 MathML Form

 α Π ( n ; z m ) z α π z 1 - α 2 α - 1 j = 0 l = 0 k = 0 ( - 1 ) j + k + l 2 - 2 j - 2 k - 2 l m l n k ( 2 j + 2 k + 2 l + 1 ) j ! l ! ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] ( 1 2 ) l TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "l"], Pochhammer] p = 0 j + k + l ( - 1 ) p ( 2 j + 2 k + 2 l + 1 p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "l"]], "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox["p", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 j + 2 k + 2 l - 2 p + 1 ) 1 F ~ 2 ( 1 ; 1 - α 2 , 3 - α 2 ; - 1 4 ( ( 2 j + 2 k + 2 l - 2 p + 1 ) 2 z 2 ) ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "l"]], "-", RowBox[List["2", " ", "p"]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], ")"]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] z α EllipticPi n z m 1 2 z 1 -1 α 2 α -1 k 0 l 0 j 0 -1 j k l 2 -2 j -1 2 k -1 2 l m l n k 2 j 2 k 2 l 1 j l -1 Pochhammer 1 2 j Pochhammer 1 2 l p 0 j k l -1 p Binomial 2 j 2 k 2 l 1 p 2 j 2 k 2 l -1 2 p 1 HypergeometricPFQRegularized 1 1 -1 α 2 -1 3 -1 α 2 -1 -1 1 4 2 j 2 k 2 l -1 2 p 1 2 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], " ", SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", "j", "+", "l"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", RowBox[List["2", " ", "j"]], "-", RowBox[List["2", " ", "l"]]]]], " ", SuperscriptBox["m", "l"], " ", SuperscriptBox["n", "k"]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "l"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["k", "+", "j", "+", "l"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]]]], ",", "p"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]], "-", RowBox[List["2", " ", "p"]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]], "-", RowBox[List["2", " ", "p"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], ")"]]]]]], "]"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]]]], ")"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["l", "!"]]]]]]]]]]]]]]]]]

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

 2001-10-29