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 EllipticPi

 http://functions.wolfram.com/08.06.06.0068.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] (-((Sqrt[1 + 1/Sqrt[1 - m]] Sqrt[Pi] n)/ (2^(5/2) (((-2 (1 + Sqrt[1 - m]) + m)/m)^(3/2) Sqrt[1 - n])))) Sum[(((-1)^q 2^(2 q + 1))/(2 q + 1)!) Sum[StirlingS2[2 q, k] Sum[(((-1)^j j! Binomial[k, k - j] n^j)/ (2^k Gamma[1/2 - k + j])) (m/(1 - Sqrt[1 - m]))^(k - j) ((1 + Sqrt[1 - n])^(-1 - j) - (1 - Sqrt[1 - n])^(-1 - j)) AppellF1[1/2, 1/2, -(3/2), 1/2 - k + j, 1/2 - 1/(2 Sqrt[1 - m]), (2 (1 + Sqrt[1 - m]))/m] z^(2 q + 1), {j, 0, k}], {k, 0, 2 q}], {q, 0, Infinity}] /; Abs[z] < 1

 Standard Form

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 MathML Form

 Π ( n ; z m ) - 2 - 5 2 π n ( m - 2 ( 1 - m + 1 ) m ) 3 / 2 1 - n 1 + 1 1 - m q = 0 ( - 1 ) q 2 2 q + 1 ( 2 q + 1 ) ! k = 0 2 q 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] 2 q ( k ) j = 0 k 2 - k ( - 1 ) j j ! n j Γ ( j - k + 1 2 ) ( k k - j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["k", "-", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( m 1 - 1 - m ) k - j ( ( 1 - n + 1 ) - j - 1 - ( 1 - 1 - n ) - j - 1 ) F 1 AppellF1 ( 1 2 ; 1 2 , - 3 2 ; j - k + 1 2 ; 1 2 - 1 2 1 - m , 2 ( 1 - m + 1 ) m ) z 2 q + 1 /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 Condition Proportional EllipticPi n z m -1 2 -1 5 2 1 2 n m -1 2 1 -1 m 1 2 1 m -1 3 2 1 -1 n 1 2 -1 1 1 1 -1 m 1 2 -1 1 2 q 0 -1 q 2 2 q 1 2 q 1 -1 k 0 2 q StirlingS2 2 q k j 0 k 2 -1 k -1 j j n j Gamma j -1 k 1 2 -1 Binomial k k -1 j m 1 -1 1 -1 m 1 2 -1 k -1 j 1 -1 n 1 2 1 -1 j -1 -1 1 -1 1 -1 n 1 2 -1 j -1 AppellF1 1 2 1 2 -1 3 2 j -1 k 1 2 1 2 -1 1 2 1 -1 m 1 2 -1 2 1 -1 m 1 2 1 m -1 z 2 q 1 z 1 [/itex]

 Rule Form

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

 2007-05-02