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; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0008.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, 2}]/Product[Gamma[Subscript[a, k]], {k, 1, 3}]) (Sum[Subscript[k, j] (1 - j)^j, {j, 0, Subscript[\[Psi], 2] - 1}] + (1 - z)^Subscript[\[Psi], 2] Sum[(Subscript[p, j] + Subscript[q, j] Log[1 - z]) (1 - z)^j, {j, 0, Infinity}]) /; Abs[1 - z] < 1 && Subscript[\[Psi], 2] == Sum[Subscript[b, j], {j, 1, 2}] - Sum[Subscript[a, j], {j, 1, 3}] && Subscript[k, j] == (((-1)^j Gamma[Subscript[a, 1] + j] Gamma[Subscript[a, 2] + j] (Subscript[\[Psi], 2] - j - 1)!)/j!) HypergeometricPFQRegularized[{Subscript[b, 1] - Subscript[a, 3], Subscript[b, 2] - Subscript[a, 3], Subscript[\[Psi], 2] - j}, {Subscript[a, 1] + Subscript[\[Psi], 2], Subscript[a, 2] + Subscript[\[Psi], 2]}, 1] && (Subscript[p, j] == ((Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], j] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], j])/ (j! (Subscript[\[Psi], 2] + j)!)) ((-1)^(Subscript[\[Psi], 2] + j) j! Sum[(Pochhammer[Subscript[b, 1] - Subscript[a, 3], k] Pochhammer[Subscript[b, 2] - Subscript[a, 3], k] (k - j - 1)!)/ (Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], k] k!), {k, j + 1, Infinity}] + (-1)^Subscript[\[Psi], 2] Sum[((Pochhammer[-j, k] Pochhammer[Subscript[b, 1] - Subscript[a, 3], k] Pochhammer[Subscript[b, 2] - Subscript[a, 3], k])/ (Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], k] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], k] k!)) (PolyGamma[j - k + 1] + PolyGamma[j + Subscript[\[Psi], 2] + 1] - PolyGamma[Subscript[\[Psi], 2] + Subscript[a, 1] + j] - PolyGamma[Subscript[\[Psi], 2] + Subscript[a, 2] + j]), {k, 0, j}]) /; Re[Subscript[a, 3]] > -j - Subscript[\[Psi], 2]) && Subscript[q, j] == (((-1)^(Subscript[\[Psi], 2] - 1) Pochhammer[Subscript[a, 1] + Subscript[\[Psi], 2], j] Pochhammer[Subscript[a, 2] + Subscript[\[Psi], 2], j])/ (j! (Subscript[\[Psi], 2] + j)!)) HypergeometricPFQRegularized[ {Subscript[b, 1] - Subscript[a, 3], Subscript[b, 2] - Subscript[a, 3], -j}, {Subscript[a, 1] + Subscript[\[Psi], 2], Subscript[a, 2] + Subscript[\[Psi], 2]}, 1] && Element[Subscript[\[Psi], 2], Integers] && Subscript[\[Psi], 2] >= 0

 Standard Form

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 MathML Form

 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] k = 1 2 Γ ( b k ) k = 1 3 Γ ( a k ) ( j = 0 ψ 2 - 1 k j ( 1 - j ) j + ( 1 - z ) ψ 2 j = 0 ( p j + q j log ( 1 - z ) ) ( 1 - z ) j ) /; "\[LeftBracketingBar]" 1 - z "\[RightBracketingBar]" < 1 ψ 2 j = 1 2 b j - j = 1 3 a j k j ( - 1 ) j Γ ( j + a 1 ) Γ ( j + a 2 ) ( ψ 2 - j - 1 ) ! j ! 3 F ~ 2 ( b 1 - a 3 , b 2 - a 3 , ψ 2 - j ; a 1 + ψ 2 , a 2 + ψ 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["\[Psi]", "2"], "-", "j"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ( p j ( a 1 + ψ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "j"], Pochhammer] ( a 2 + ψ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "j"], Pochhammer] j ! ( j + ψ 2 ) ! ( ( - 1 ) j + ψ 2 j ! k = j + 1 ( b 1 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( b 2 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( k - j - 1 ) ! ( a 1 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] ( a 2 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] k ! + ( - 1 ) ψ 2 k = 0 j ( - j ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "j"]], ")"]], "k"], Pochhammer] ( b 1 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( b 2 - a 3 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], ")"]], "k"], Pochhammer] ( a 1 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] ( a 2 + ψ 2 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "k"], Pochhammer] k ! ( ψ TagBox["\[Psi]", PolyGamma] ( j - k + 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( j + ψ 2 + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 1 + ψ 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( j + a 2 + ψ 2 ) ) ) /; Re ( a 3 ) > - j - ψ 2 ) q j ( - 1 ) ψ 2 - 1 ( a 1 + ψ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "j"], Pochhammer] ( a 2 + ψ 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], ")"]], "j"], Pochhammer] j ! ( j + ψ 2 ) ! 3 F ~ 2 ( b 1 - a 3 , b 2 - a 3 , - j ; a 1 + ψ 2 , a 2 + ψ 2 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["-", "j"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", SubscriptBox["\[Psi]", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] ψ 2 TagBox["\[DoubleStruckCapitalN]", Function[Integers]] Condition HypergeometricPFQ Subscript a 1 Subscript a 2 Subscript a 3 Subscript b 1 Subscript b 2 z k 1 2 Gamma Subscript b k k 1 3 Gamma Subscript a k -1 j 0 Subscript ψ 2 -1 Subscript k j 1 -1 j j 1 -1 z Subscript ψ 2 j 0 Subscript p j Subscript q j 1 -1 z 1 -1 z j 1 -1 z 1 Subscript ψ 2 j 1 2 Subscript b j -1 j 1 3 Subscript a j Subscript k j -1 j Gamma j Subscript a 1 Gamma j Subscript a 2 Subscript ψ 2 -1 j -1 j -1 HypergeometricPFQRegularized Subscript b 1 -1 Subscript a 3 Subscript b 2 -1 Subscript a 3 Subscript ψ 2 -1 j Subscript a 1 Subscript ψ 2 Subscript a 2 Subscript ψ 2 1 Condition Subscript p j Pochhammer Subscript a 1 Subscript ψ 2 j Pochhammer Subscript a 2 Subscript ψ 2 j j j Subscript ψ 2 -1 -1 j Subscript ψ 2 j k j 1 Pochhammer Subscript b 1 -1 Subscript a 3 k Pochhammer Subscript b 2 -1 Subscript a 3 k k -1 j -1 Pochhammer Subscript a 1 Subscript ψ 2 k Pochhammer Subscript a 2 Subscript ψ 2 k k -1 -1 Subscript ψ 2 k 0 j Pochhammer -1 j k Pochhammer Subscript b 1 -1 Subscript a 3 k Pochhammer Subscript b 2 -1 Subscript a 3 k Pochhammer Subscript a 1 Subscript ψ 2 k Pochhammer Subscript a 2 Subscript ψ 2 k k -1 PolyGamma j -1 k 1 PolyGamma j Subscript ψ 2 1 -1 PolyGamma j Subscript a 1 Subscript ψ 2 -1 PolyGamma j Subscript a 2 Subscript ψ 2 Subscript a 3 -1 j -1 Subscript ψ 2 Subscript q j -1 Subscript ψ 2 -1 Pochhammer Subscript a 1 Subscript ψ 2 j Pochhammer Subscript a 2 Subscript ψ 2 j j j Subscript ψ 2 -1 HypergeometricPFQRegularized Subscript b 1 -1 Subscript a 3 Subscript b 2 -1 Subscript a 3 -1 j Subscript a 1 Subscript ψ 2 Subscript a 2 Subscript ψ 2 1 Subscript ψ 2 [/itex]

 Rule Form

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

 2001-10-29