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; }

 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.07.0006.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], â€¦, Subscript[a, p]}, {Subscript[b, 1], â€¦, Subscript[b, p], Subscript[b, p + 1], â€¦, Subscript[b, q]}, z] == Product[(1/(Gamma[Subscript[a, k]]*Gamma[Subscript[b, k] - Subscript[a, k]]))* Integrate[Product[Subscript[t, k]^(Subscript[a, k] - 1)* (1 - Subscript[t, k])^ (Subscript[b, k] - Subscript[a, k] - 1)* HypergeometricPFQRegularized[{}, {Subscript[b, p + 1], â€¦, Subscript[b, q]}, z*Product[Subscript[t, k], {k, 1, p}]], {k, 1, p}], {Subscript[t, 1], 0, 1}, {Subscript[t, 2], 0, 1}, â€¦, {Subscript[t, p], 0, 1}], {k, 1, p}] /; Re[Subscript[b, k]] > Re[Subscript[a, k]] > 0 && 1 <= k <= p

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "p"], ",", SubscriptBox["b", RowBox[List["p", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[FractionBox[RowBox[List[" ", "1"]], RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "k"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "k"], "-", SubscriptBox["a", "k"]]], "]"]]]]]]], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List["\[Ellipsis]", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], RowBox[List[SubsuperscriptBox["t", "k", RowBox[List[SubscriptBox["a", "k"], "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["t", "k"]]], ")"]], RowBox[List[SubscriptBox["b", "k"], "-", SubscriptBox["a", "k"], "-", "1"]]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["p", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["t", "k"]]]]]]], "]"]]]]]], ")"]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "1"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "2"]]], " ", "\[Ellipsis]", RowBox[List["\[DifferentialD]", SubscriptBox["t", "p"]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", SubscriptBox["b", "k"], "]"]], ">", RowBox[List["Re", "[", SubscriptBox["a", "k"], "]"]], ">", "0"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "p"]]]]]]]]

 MathML Form

 p F ~ q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "p"], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", "z"]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] ( k = 1 p 1 Γ ( a k ) Γ ( b k - a k ) ) 0 1 0 1 k = 1 p t k a k - 1 ( 1 - t k ) b k - a k - 1 0 F ~ q - p ( ; b p + 1 , , b q ; z k = 1 p t k ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["q", "-", "p"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["", HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", RowBox[List["p", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["t", "k"]]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] t 1 t p /; Re ( b k ) > Re ( a k ) > 0 1 k p p F ~ q ( a 1 , , a p ; b 1 , , b q ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "p"], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", "z"]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] ( k = 1 p 1 Γ ( a k ) Γ ( b k - a k ) ) 0 1 0 1 k = 1 p t k a k - 1 ( 1 - t k ) b k - a k - 1 0 F ~ q - p ( ; b p + 1 , , b q ; z k = 1 p t k ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox[OverscriptBox["F", "~"], FormBox[RowBox[List["q", "-", "p"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["", HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", RowBox[List["p", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "p"], SubscriptBox["t", "k"]]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]] t 1 t p /; Re ( b k ) > Re ( a k ) > 0 1 k p [/itex]

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

 2001-10-29