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 EllipticPi

 http://functions.wolfram.com/08.06.06.0071.01

 Input Form

 EllipticPi[n, z, m] == (1/Sqrt[m]) EllipticPi[n/m, 1/m] + 2 u EllipticPi[n, m] - ((m Sqrt[2])/((n - m) Sqrt[-1 + m])) Sqrt[(-Sqrt[-1 + m]) (z - Subscript[z, 0])] Sum[(1/(2 k + 1)) Sum[(j + 1) Subscript[c, k - j] Sum[(((-1)^r Binomial[j, r])/(1 + r)) Subscript[q, r, j] (z - Subscript[z, 0])^k, {r, 0, j}], {j, 0, k}], {k, 0, Infinity}] /; Subscript[z, 0] == ArcCsc[Sqrt[m]] + Pi u && Element[u, Integers] && Subscript[a, 0] == 1 && Subscript[a, 2 k] == ((-1)^k 2^(2 k))/(2 k + 1)! && Subscript[a, 2 k + 1] == ((-1)^(k - 1) 2^(2 k) (2 - m))/ (Sqrt[m - 1] (2 k + 2)!) && Subscript[b, 0] == 1 && Subscript[b, k] == ((2 n Sqrt[-1 + m])/(n - m)) Subscript[a, k - 1] && Subscript[c, k] == (Pochhammer[3/2, k]/k!) Sum[(((-1)^j Binomial[k, j])/(2 j + 1)) Subscript[p, j, k], {j, 0, k}] && Subscript[p, u, 0] == 1 && Subscript[p, u, v] == (1/v) Sum[(u j - v + j) Subscript[a, j] Subscript[p, u, v - j], {j, 1, v}] && Subscript[q, u, 0] == 1 && Subscript[q, j, k] == (1/k) Sum[(j i - k + i) Subscript[b, i] Subscript[q, j, k - i], {i, 1, k}] && Element[k, Integers] && k >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", SqrtBox["m"]], RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]]]], "+", RowBox[List["2", "u", " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["m", SqrtBox["2"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]]], SqrtBox[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], SubscriptBox["c", 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["q", RowBox[List["r", ",", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "k"]]]]]]]]]]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List[SubscriptBox["z", "0"], "\[Equal]", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List["2", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["2", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "!"]]]]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], SuperscriptBox["2", RowBox[List["2", "k"]]], " ", RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "2"]], ")"]], "!"]]]]]]], "\[And]", RowBox[List[SubscriptBox["b", "0"], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", " ", "n", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], RowBox[List["n", "-", "m"]]], SubscriptBox["a", RowBox[List["k", "-", "1"]]]]]]], "\[And]", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], RowBox[List["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"]], "+", "1"]]], SubscriptBox["p", RowBox[List["j", ",", "k"]]]]]]]]]]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "v"]]], "\[Equal]", RowBox[List[FractionBox["1", "v"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "v"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["u", " ", "j"]], "-", "v", "+", "j"]], ")"]], SubscriptBox["a", "j"], " ", SubscriptBox["p", RowBox[List["u", ",", RowBox[List["v", "-", "j"]]]]]]]]]]]]], "\[And]", RowBox[List[SubscriptBox["q", RowBox[List["u", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["q", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "i"]], "-", "k", "+", "i"]], ")"]], " ", SubscriptBox["b", "i"], SubscriptBox["q", RowBox[List["j", ",", RowBox[List["k", "-", "i"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 Π ( n ; z m ) 2 u Π ( n m ) + 1 m Π ( n m 1 m ) - m 2 - m - 1 ( z - z 0 ) ( n - m ) m - 1 k = 0 1 2 k + 1 j = 0 k ( j + 1 ) c 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 q r , j ( z - z 0 ) k /; z 0 π u + csc - 1 ( m ) u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] a 0 1 a 2 k ( - 1 ) k 2 2 k ( 2 k + 1 ) ! a 2 k + 1 ( - 1 ) k - 1 2 2 k ( 2 - m ) m - 1 ( 2 k + 2 ) ! b 0 1 b k 2 n m - 1 n - m a k - 1 c k 1 k ! ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] j = 0 k ( - 1 ) j 2 j + 1 ( 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 p u , 0 1 p u , v 1 v j = 1 v ( u j + j - v ) a j p u , v - j q u , 0 1 q j , k 1 k i = 1 k ( j i + i - k ) b i q j , k - i k Condition EllipticPi n z m 2 u EllipticPi n m 1 m 1 2 -1 EllipticPi n m -1 1 m -1 -1 m 2 1 2 -1 m -1 1 2 z -1 Subscript z 0 1 2 n -1 m m -1 1 2 -1 k 0 1 2 k 1 -1 j 0 k j 1 Subscript c k -1 j r 0 j -1 r Binomial j r r 1 -1 Subscript q r j z -1 Subscript z 0 k Subscript z 0 u m 1 2 u Subscript a 0 1 Subscript a 2 k -1 k 2 2 k 2 k 1 -1 Subscript a 2 k 1 -1 k -1 2 2 k 2 -1 m m -1 1 2 2 k 2 -1 Subscript b 0 1 Subscript b k 2 n m -1 1 2 n -1 m -1 Subscript a k -1 Subscript c k 1 k -1 Pochhammer 3 2 k j 0 k -1 j 2 j 1 -1 Binomial k j Subscript p j k Subscript p u 0 1 Subscript p u v 1 v -1 j 1 v u j j -1 v Subscript a j Subscript p u v -1 j Subscript q u 0 1 Subscript q j k 1 k -1 i 1 k j i i -1 k Subscript b i Subscript q j k -1 i k [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]], SqrtBox["m"]], "+", RowBox[List["2", " ", "u", " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", " ", SqrtBox["2"]]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "1"]], ")"]], " ", SubscriptBox["c", 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["q", RowBox[List["r", ",", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "k"]]], RowBox[List["1", "+", "r"]]]]]]]]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["zz", "0"], "\[Equal]", RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["a", RowBox[List["2", " ", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", RowBox[List["(", RowBox[List["2", "-", "m"]], ")"]]]], RowBox[List[SqrtBox[RowBox[List["m", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "2"]], ")"]], "!"]]]]]]], "&&", RowBox[List[SubscriptBox["b", "0"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "m"]]]]], ")"]], " ", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], RowBox[List["n", "-", "m"]]]]], "&&", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[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"]]]]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]]]]], RowBox[List["k", "!"]]]]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["u", ",", "v"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "v"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["u", " ", "j"]], "-", "v", "+", "j"]], ")"]], " ", SubscriptBox["a", "j"], " ", SubscriptBox["p", RowBox[List["u", ",", RowBox[List["v", "-", "j"]]]]]]]]], "v"]]], "&&", RowBox[List[SubscriptBox["q", RowBox[List["u", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["q", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "i"]], "-", "k", "+", "i"]], ")"]], " ", SubscriptBox["b", "i"], " ", SubscriptBox["q", RowBox[List["j", ",", RowBox[List["k", "-", "i"]]]]]]]]], "k"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02