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 Gamma

 http://functions.wolfram.com/06.07.20.0008.02

 Input Form

 D[Gamma[a, Subscript[z, 1], Subscript[z, 2]], {a, n}] == Subscript[z, 2]^a Sum[(-1)^(n - j) Binomial[n, j] (n - j)! Gamma[a]^(n - j + 1) Log[Subscript[z, 2]]^j HypergeometricPFQRegularized[{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, n - j + 1]}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, n - j + 1]}, -Subscript[z, 2]], {j, 0, n}] - Subscript[z, 1]^a Sum[(-1)^(n - j) Binomial[n, j] (n - j)! Gamma[a]^(n - j + 1) Log[Subscript[z, 1]]^j HypergeometricPFQRegularized[ {Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, n - j + 1]}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, n - j + 1]}, -Subscript[z, 1]], {j, 0, n}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == a && 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["Gamma", "[", RowBox[List["a", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[SubsuperscriptBox["z", "2", "a"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["n", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SubscriptBox["z", "2"]]]]], "]"]]]]]]]], "-", RowBox[List[SubsuperscriptBox["z", "1", "a"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["n", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SubscriptBox["z", "1"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "a"]], "\[And]", " ", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n Γ ( a , z 1 , z 2 ) a n z 2 a j = 0 n ( - 1 ) n - j ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - j ) ! Γ ( a ) n - j + 1 log j ( z 2 ) n - j + 1 F ~ n - j + 1 ( a 1 , a 2 , , a n - j + 1 ; a 1 + 1 , a 2 + 1 , , a n - j + 1 + 1 ; - z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", RowBox[List["n", "-", "j", "+", "1"]]], SubscriptBox[OverscriptBox["F", "~"], RowBox[List["n", "-", "j", "+", "1"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[RowBox[List["-", SubscriptBox["z", "2"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] - z 1 a j = 0 n ( - 1 ) n - j ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - j ) ! Γ ( a ) n - j + 1 log j ( z 1 ) n - j + 1 F ~ n - j + 1 ( a 1 , a 2 , , a n - j + 1 ; a 1 + 1 , a 2 + 1 , , a n - j + 1 + 1 ; - z 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", RowBox[List["n", "-", "j", "+", "1"]]], SubscriptBox[OverscriptBox["F", "~"], RowBox[List["n", "-", "j", "+", "1"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]], "+", "1"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[RowBox[List["-", SubscriptBox["z", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] /; a 1 a 2 a n + 1 a n Condition a n Gamma a Subscript z 1 Subscript z 2 Subscript z 2 a j 0 n -1 n -1 j Binomial n j n -1 j Gamma a n -1 j 1 Subscript z 2 j HypergeometricPFQRegularized Subscript a 1 Subscript a 2 Subscript a n -1 j 1 Subscript a 1 1 Subscript a 2 1 Subscript a n -1 j 1 1 -1 Subscript z 2 -1 Subscript z 1 a j 0 n -1 n -1 j Binomial n j n -1 j Gamma a n -1 j 1 Subscript z 1 j HypergeometricPFQRegularized Subscript a 1 Subscript a 2 Subscript a n -1 j 1 Subscript a 1 1 Subscript a 2 1 Subscript a n -1 j 1 1 -1 Subscript z 1 Subscript a 1 Subscript a 2 Subscript a n 1 a 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["Gamma", "[", RowBox[List["a_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["zz", "2", "a"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["n", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SubscriptBox["zz", "2"]]]]], "]"]]]]]]]], "-", RowBox[List[SubsuperscriptBox["zz", "1", "a"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["n", "-", "j", "+", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]], "j"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["n", "-", "j", "+", "1"]]]]]]], "}"]], ",", RowBox[List["-", SubscriptBox["zz", "1"]]]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["aa", "1"], "\[Equal]", SubscriptBox["aa", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "a"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2001-10-29