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; }

 Hypergeometric2F1Regularized

 http://functions.wolfram.com/07.24.06.0065.01

 Input Form

 Hypergeometric2F1Regularized[a, a + n, c, z] == Subscript[F, Infinity][z, a, a + n, c] /; (Subscript[F, n][z, a, a + n, c] == ((-z)^(-a - n)/(Gamma[a + n] Gamma[c - a])) Sum[(((Pochhammer[a, k + n] Pochhammer[1 + a - c, k + n])/ (k! (k + n)!)) (PolyGamma[k + n + 1] + PolyGamma[k + 1] - PolyGamma[a + n + k] - PolyGamma[c - a - n - k] + Log[-z]))/z^k, {k, 0, m}] + (1/((-z)^a Gamma[a + n])) Sum[(Pochhammer[a, k] Gamma[n - k])/(k! Gamma[c - a - k])/z^k, {k, 0, n - 1}] == Hypergeometric2F1Regularized[a, a + n, c, z] - ((-1)^n/(Gamma[a] Gamma[a + n])) MeijerG[{{-a - m - n, -a - m - n}, {1 - a, 1 - a - n}}, {{0, -a - n - m, -a - n - m}, {1 - c}}, -z] && Element[m, Integers] && m >= 0) && Element[n, Integers] && n >= 0 && !Element[c - a, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["a", "+", "n"]], ",", "c", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "a", ",", RowBox[List["a", "+", "n"]], ",", "c"]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", "a", ",", RowBox[List["a", "+", "n"]], ",", "c"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "n"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", RowBox[List["k", "+", "n"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "c"]], ",", RowBox[List["k", "+", "n"]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "n", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "n", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["c", "-", "a", "-", "n", "-", "k"]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]]]], ")"]], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["n", "-", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List["Gamma", "[", RowBox[List["c", "-", "a", "-", "k"]], "]"]]]]], SuperscriptBox["z", RowBox[List["-", "k"]]]]]]]]]]], "\[Equal]", RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["a", "+", "n"]], ",", "c", ",", "z"]], "]"]], "-", " ", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["1", "-", "a", "-", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[RowBox[List["-", "a"]], "-", "n", "-", "m"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "n", "-", "m"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "c"]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List["c", "-", "a"]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 2 F ~ 1 ( a , a + n ; c ; z ) 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["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] F ( z , a , a + n , c ) /; ( ( F n ( z , a , a + n , c ) ( - z ) - a Γ ( a + n ) k = 0 n - 1 ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] Γ ( n - k ) z - k k ! Γ ( - a + c - k ) + ( - z ) - a - n Γ ( a + n ) Γ ( c - a ) k = 0 m ( a ) k + n TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] ( a - c + 1 ) k + n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "c", "+", "1"]], ")"]], RowBox[List["k", "+", "n"]]], Pochhammer] k ! ( k + n ) ! ( log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( k + n + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( - a + c - k - n ) - ψ TagBox["\[Psi]", PolyGamma] ( a + k + n ) ) z - k 2 F ~ 1 ( a , a + n ; c ; z ) 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["a", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "n"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] - ( - 1 ) n Γ ( a ) Γ ( a + n ) G 4 , 4 3 , 2 ( - z - a - m - n , - a - m - n , 1 - a , - a - n + 1 0 , - a - m - n , - a - m - n , 1 - c ) TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "4"]], RowBox[List["3", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "z"]], MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "a"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "n", "+", "1"]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", "c"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] ) m ) n ¬ c - a TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition Hypergeometric2F1Regularized a a n c z Subscript F z a a n c Subscript F n z a a n c -1 z -1 a Gamma a n -1 k 0 n -1 Pochhammer a k Gamma n -1 k z -1 k k Gamma -1 a c -1 k -1 -1 z -1 a -1 n Gamma a n Gamma c -1 a -1 k 0 m Pochhammer a k n Pochhammer a -1 c 1 k n k k n -1 -1 z PolyGamma k 1 PolyGamma k n 1 -1 PolyGamma -1 a c -1 k -1 n -1 PolyGamma a k n z -1 k Hypergeometric2F1Regularized a a n c z -1 -1 n Gamma a Gamma a n -1 MeijerG -1 a -1 m -1 n -1 a -1 m -1 n 1 -1 a -1 a -1 n 1 0 -1 a -1 m -1 n -1 a -1 m -1 n 1 -1 c -1 z m n c -1 a [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a_", ",", RowBox[List["a_", "+", "n_"]], ",", "c_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "a", ",", RowBox[List["a", "+", "n"]], ",", "c"]], "]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", "a", ",", RowBox[List["a", "+", "n"]], ",", "c"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "a"]], "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "m"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", RowBox[List["k", "+", "n"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "c"]], ",", RowBox[List["k", "+", "n"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "n", "+", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "n", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["c", "-", "a", "-", "n", "-", "k"]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "n"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "a"]], "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", "a"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "-", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["c", "-", "a", "-", "k"]], "]"]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]], "\[Equal]", RowBox[List[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["a", ",", RowBox[List["a", "+", "n"]], ",", "c", ",", "z"]], "]"]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "m", "-", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["1", "-", "a", "-", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[RowBox[List["-", "a"]], "-", "n", "-", "m"]], ",", RowBox[List[RowBox[List["-", "a"]], "-", "n", "-", "m"]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "c"]], "}"]]]], "}"]], ",", RowBox[List["-", "z"]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "n"]], "]"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List[RowBox[List["c", "-", "a"]], "\[Element]", "Integers"]]]]]]]]]]]]

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

 2007-05-02