html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArithmeticGeometricMean

 http://functions.wolfram.com/09.54.20.0005.01

 Input Form

 D[ArithmeticGeometricMean[a, b], {a, n}] == ArithmeticGeometricMean[a, b] KroneckerDelta[n] + (Pi/(4 b^n)) ((b KroneckerDelta[n - 1])/EllipticK[((a - b)/(a + b))^2] + b n n! Sum[((-1)^q/((q + 1)! (n - q - 1)!)) EllipticK[((a - b)/(a + b))^2]^(-q - 1) Sum[\[Ellipsis] Sum[Product[Binomial[n - 1 - Sum[Subscript[k, j], {j, 1, p - 1}], Subscript[k, p]], {p, 1, q - 1}] Product[A[Subscript[k, i], a, b], {i, 1, q - 1}] A[n - 1 - Sum[Subscript[k, j], {j, 1, q - 1}], a, b], {Subscript[k, q - 1], 0, n - 1 - Sum[Subscript[k, j], {j, 1, p}]}], {Subscript[k, 1], 0, n - 1 - Sum[Subscript[k, j], {j, 1, p}]}, {Subscript[k, 2], 0, n - 1 - Sum[Subscript[k, j], {j, 1, p}]}], {q, 1, n - 1}] + (a + b) (n + 1)! Sum[((-1)^q/((q + 1)! (n - q)!)) EllipticK[((a - b)/(a + b))^2]^ (-q - 1) Sum[\[Ellipsis] Sum[Product[Binomial[n - Sum[Subscript[k, j], {j, 1, p - 1}], Subscript[k, p]], {p, 1, q - 1}] Product[A[Subscript[k, i], a, b], {i, 1, q - 1}] A[n - Sum[Subscript[k, j], {j, 1, q - 1}], a, b], {Subscript[k, q - 1], 0, n - Sum[Subscript[k, j], {j, 1, p}]}], {Subscript[k, 1], 0, n - Sum[Subscript[k, j], {j, 1, p}]}, {Subscript[k, 2], 0, n - Sum[Subscript[k, j], {j, 1, p}]}], {q, 1, n}]) /; A[r, a, b] == EllipticK[((a - b)/(a + b))^2] KroneckerDelta[r] + (Pi/2) Sum[(1/m!) Sum[(Pochhammer[2 s - m + 1, 2 (m - s)]/ ((m - s)! 2^(m - 2 s))) ((a + b)/(a - b))^m Hypergeometric2F1Regularized[1/2, 1/2, 1 - s, ((a - b)/(a + b))^2] Sum[(-1)^q Binomial[m, q] ((a - b)/(a + b))^q Sum[\[Ellipsis] Sum[KroneckerDelta[r, Sum[Subscript[u, i], {i, 1, m - q}]] Multinomial[Subscript[u, 1], Subscript[u, 2], \[Ellipsis], Subscript[u, m - q]] Product[ KroneckerDelta[Subscript[u, i]] - (2 (-1)^Subscript[u, i] b^(Subscript[u, i] + 1) Subscript[u, i]!)/(a + b)^ (Subscript[u, i] + 1), {i, 1, m - q}], {Subscript[u, m - q], 0, r}], {Subscript[u, 1], 0, r}, {Subscript[u, 2], 0, r}], {q, 0, m}], {s, 0, m}], {m, 1, r}] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["a", ",", "n"]], "}"]]], RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "b"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "b"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", RowBox[List[FractionBox["\[Pi]", RowBox[List["4", SuperscriptBox["b", "n"]]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List["b", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]], RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]]], "+", RowBox[List["b", " ", "n", " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "q", "-", "1"]], ")"]], "!"]]]]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]], RowBox[List[RowBox[List["-", "q"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["q", "-", "1"]]], "=", "0"]], RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["p", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", SubscriptBox["k", "p"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["A", "[", RowBox[List[SubscriptBox["k", "i"], ",", "a", ",", "b"]], "]"]]]], ")"]], " ", RowBox[List["A", "[", RowBox[List[RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["q", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", "a", ",", "b"]], "]"]]]]]]]]]]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], "n"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "q"]], ")"]], "!"]]]]], " ", SuperscriptBox[RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]], RowBox[List[RowBox[List["-", "q"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["q", "-", "1"]]], "=", "0"]], RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["p", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", SubscriptBox["k", "p"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["A", "[", RowBox[List[SubscriptBox["k", "i"], ",", "a", ",", "b"]], "]"]]]], ")"]], " ", RowBox[List["A", "[", RowBox[List[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["q", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", "a", ",", "b"]], "]"]]]]]]]]]]]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["A", "[", RowBox[List["r", ",", "a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]], " ", RowBox[List["KroneckerDelta", "[", "r", "]"]]]], "+", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "r"], RowBox[List[FractionBox["1", RowBox[List["m", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "s"]], "-", "m", "+", "1"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "s"]], ")"]]]]]], "]"]], " "]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "s"]], ")"]], "!"]], " ", SuperscriptBox["2", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]]], " "]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["a", "-", "b"]]], ")"]], "m"], RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["1", "-", "s"]], ",", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"]]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "q"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "q"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["u", "1"], "=", "0"]], "r"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["u", "2"], "=", "0"]], "r"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["u", RowBox[List["m", "-", "q"]]], "=", "0"]], "r"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["r", ",", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["m", "-", "q"]]], SubscriptBox["u", "i"]]]]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["u", "1"], ",", SubscriptBox["u", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["u", RowBox[List["m", "-", "q"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], RowBox[List["m", "-", "q"]]], RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", SubscriptBox["u", "i"], "]"]], "-", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["u", "i"]], " ", SuperscriptBox["b", RowBox[List[SubscriptBox["u", "i"], "+", "1"]]], RowBox[List[SubscriptBox["u", "i"], "!"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List[SubscriptBox["u", "i"], "+", "1"]]]]]], ")"]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n agm ArithmeticGeometricMean ( a , b ) a n agm ArithmeticGeometricMean ( a , b ) δ KroneckerDelta n + π 4 b n ( b δ KroneckerDelta n - 1 K ( ( a - b a + b ) 2 ) + b n n ! q = 1 n - 1 ( - 1 ) q ( q + 1 ) ! ( n - q - 1 ) ! K ( ( a - b a + b ) 2 ) - q - 1 k 1 = 0 n - j = 1 p k j - 1 k 2 = 0 n - j = 1 p k j - 1 k q - 1 = 0 n - j = 1 p k j - 1 ( p = 1 q - 1 ( n - j = 1 p - 1 k j - 1 k p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "p"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ) ( i = 1 q - 1 A ( k i , a , b ) ) A ( n - j = 1 q - 1 k j - 1 , a , b ) + ( a + b ) ( n + 1 ) ! q = 1 n ( - 1 ) q ( q + 1 ) ! ( n - q ) ! K ( ( a - b a + b ) 2 ) - q - 1 k 1 = 0 n - j = 1 p k j k 2 = 0 n - j = 1 p k j k q - 1 = 0 n - j = 1 p k j ( p = 1 q - 1 ( n - j = 1 p - 1 k j k p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]]]], Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "p"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ) ( i = 1 q - 1 A ( k i , a , b ) ) A ( n - j = 1 q - 1 k j , a , b ) ) /; A ( r , a , b ) K ( ( a - b a + b ) 2 ) δ KroneckerDelta r + π 2 m = 1 r 1 m ! s = 0 m 1 ( m - s ) ! 2 m - 2 s ( ( 2 s - m + 1 ) 2 ( m - s ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "m", "+", "1"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "s"]], ")"]]]]], Pochhammer] ( a + b a - b ) m 2 F ~ 1 ( 1 2 , 1 2 ; 1 - s ; ( a - b a + b ) 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "s"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] q = 0 m ( - 1 ) q ( m q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( a - b a + b ) q u 1 = 0 r u 2 = 0 r u m - q = 0 r δ KroneckerDelta r , i = 1 m - q u i ( u 1 + u 2 + + u m - q ; u 1 , u 2 , , u m - q ) i = 1 m - q ( δ KroneckerDelta u i - 2 ( - 1 ) u i b u i + 1 u i ! ( a + b ) u i + 1 ) ) /; n n agm ArithmeticGeometricMean ( a , b ) a n agm ArithmeticGeometricMean ( a , b ) δ KroneckerDelta n + π 4 b n ( b δ KroneckerDelta n - 1 K ( ( a - b a + b ) 2 ) + b n n ! q = 1 n - 1 ( - 1 ) q ( q + 1 ) ! ( n - q - 1 ) ! K ( ( a - b a + b ) 2 ) - q - 1 k 1 = 0 n - j = 1 p k j - 1 k 2 = 0 n - j = 1 p k j - 1 k q - 1 = 0 n - j = 1 p k j - 1 ( p = 1 q - 1 ( n - j = 1 p - 1 k j - 1 k p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "p"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ) ( i = 1 q - 1 A ( k i , a , b ) ) A ( n - j = 1 q - 1 k j - 1 , a , b ) + ( a + b ) ( n + 1 ) ! q = 1 n ( - 1 ) q ( q + 1 ) ! ( n - q ) ! K ( ( a - b a + b ) 2 ) - q - 1 k 1 = 0 n - j = 1 p k j k 2 = 0 n - j = 1 p k j k q - 1 = 0 n - j = 1 p k j ( p = 1 q - 1 ( n - j = 1 p - 1 k j k p ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]]]], Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "p"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ) ( i = 1 q - 1 A ( k i , a , b ) ) A ( n - j = 1 q - 1 k j , a , b ) ) /; A ( r , a , b ) K ( ( a - b a + b ) 2 ) δ KroneckerDelta r + π 2 m = 1 r 1 m ! s = 0 m 1 ( m - s ) ! 2 m - 2 s ( ( 2 s - m + 1 ) 2 ( m - s ) TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "m", "+", "1"]], ")"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "s"]], ")"]]]]], Pochhammer] ( a + b a - b ) m 2 F ~ 1 ( 1 2 , 1 2 ; 1 - s ; ( a - b a + b ) 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["1", "-", "s"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] q = 0 m ( - 1 ) q ( m q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( a - b a + b ) q u 1 = 0 r u 2 = 0 r u m - q = 0 r δ KroneckerDelta r , i = 1 m - q u i ( u 1 + u 2 + + u m - q ; u 1 , u 2 , , u m - q ) i = 1 m - q ( δ KroneckerDelta u i - 2 ( - 1 ) u i b u i + 1 u i ! ( a + b ) u i + 1 ) ) /; n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", "n_"]], "}"]]]]], RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "b"]], "]"]], " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["b", " ", RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]], RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]]], "+", RowBox[List["b", " ", "n", " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]], RowBox[List[RowBox[List["-", "q"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["q", "-", "1"]]], "=", "0"]], RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["p", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", SubscriptBox["k", "p"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["A", "[", RowBox[List[SubscriptBox["k", "i"], ",", "a", ",", "b"]], "]"]]]], ")"]], " ", RowBox[List["A", "[", RowBox[List[RowBox[List["n", "-", "1", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["q", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", "a", ",", "b"]], "]"]]]]]]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "q", "-", "1"]], ")"]], "!"]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", SuperscriptBox[RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]], RowBox[List[RowBox[List["-", "q"]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "1"], "=", "0"]], RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", "2"], "=", "0"]], RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["q", "-", "1"]]], "=", "0"]], RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "p"], SubscriptBox["k", "j"]]]]]], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["p", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["p", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", SubscriptBox["k", "p"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List["A", "[", RowBox[List[SubscriptBox["k", "i"], ",", "a", ",", "b"]], "]"]]]], ")"]], " ", RowBox[List["A", "[", RowBox[List[RowBox[List["n", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["q", "-", "1"]]], SubscriptBox["k", "j"]]]]], ",", "a", ",", "b"]], "]"]]]]]]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["q", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "q"]], ")"]], "!"]]]]]]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["b", "n"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["A", "[", RowBox[List["r", ",", "a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["EllipticK", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"], "]"]], " ", RowBox[List["KroneckerDelta", "[", "r", "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "r"], FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "s"]], "-", "m", "+", "1"]], ",", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "s"]], ")"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["a", "-", "b"]]], ")"]], "m"], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["1", "-", "s"]], ",", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "m"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "q"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "q"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["u", "1"], "=", "0"]], "r"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["u", "2"], "=", "0"]], "r"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["u", RowBox[List["m", "-", "q"]]], "=", "0"]], "r"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List["r", ",", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], RowBox[List["m", "-", "q"]]], SubscriptBox["u", "i"]]]]], "]"]], " ", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["u", "1"], ",", SubscriptBox["u", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["u", RowBox[List["m", "-", "q"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], RowBox[List["m", "-", "q"]]], RowBox[List["(", RowBox[List[RowBox[List["KroneckerDelta", "[", SubscriptBox["u", "i"], "]"]], "-", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], SubscriptBox["u", "i"]], " ", SuperscriptBox["b", RowBox[List[SubscriptBox["u", "i"], "+", "1"]]], " ", RowBox[List[SubscriptBox["u", "i"], "!"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], RowBox[List[SubscriptBox["u", "i"], "+", "1"]]]]]], ")"]]]]]]]]]]]]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "s"]], ")"]], "!"]], " ", SuperscriptBox["2", RowBox[List["m", "-", RowBox[List["2", " ", "s"]]]]]]]]]], RowBox[List["m", "!"]]]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02