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 Zeta

 http://functions.wolfram.com/10.02.20.0028.01

 Input Form

 Derivative[1, 0][Zeta][-5, a] == 120 PolyGamma[-6, -1 + a] + (36697 + 42 a (-4220 + a (8275 - 822 a (10 + (-5 + a) a))))/15120 + (1/252) (1 + 21 (-1 + a)^2 a^2 (-1 + 2 (-1 + a) a)) EulerGamma + (-1 + a)^5 Log[-1 + a] - 5 (-1 + a)^4 Log[Glaisher] - (1/2) (-1 + a)^5 Log[2 Pi] - (5 (-1 + a) (2 (-1 + a)^2 Pi^2 Zeta[3] - 3 Zeta[5]))/(4 Pi^4) + (-(137/60) + EulerGamma) Zeta[-5, a] + Derivative[1][Zeta][-5] + 10 (-1 + a)^2 Derivative[1][Zeta][-3] + Sum[(a + k)^5 Log[a + k] - (1/2) ((a + k)^2)^(5/2) Log[(a + k)^2], {k, 0, Floor[1 - Re[a]]}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "5"]], ",", "a"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["120", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "6"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "a"]]]], "]"]]]], "+", FractionBox[RowBox[List["36697", "+", RowBox[List["42", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4220"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["8275", "-", RowBox[List["822", " ", "a", " ", RowBox[List["(", RowBox[List["10", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "a"]], ")"]], " ", "a"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "15120"], "+", RowBox[List[FractionBox["1", "252"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["21", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "2"], " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", "a"]]]], ")"]]]]]], ")"]], " ", "EulerGamma"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "5"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], "]"]]]], "-", RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "4"], " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "5"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Zeta", "[", "5", "]"]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["137", "60"]]], "+", "EulerGamma"]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "5"]], ",", "a"]], "]"]]]], "+", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]], "+", RowBox[List["10", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "2"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["1", "-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "5"], " ", RowBox[List["Log", "[", RowBox[List["a", "+", "k"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], "]"]]]]]], ")"]]]]]]]]]]

 MathML Form

 ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( - 5 , a ) 120 ψ TagBox["\[Psi]", PolyGamma] ( - 6 ) ( a - 1 ) + log ( a - 1 ) ( a - 1 ) 5 - 1 2 log ( 2 π ) ( a - 1 ) 5 - 5 log ( A TagBox["A", Function[List[], Glaisher]] ) ( a - 1 ) 4 + 10 Zeta ( - 3 ) ( a - 1 ) 2 - 5 ( 2 ( a - 1 ) 2 π 2 ζ ( 3 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["3", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] - 3 ζ ( 5 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["5", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] ) ( a - 1 ) 4 π 4 + 42 a ( a ( 8275 - 822 a ( ( a - 5 ) a + 10 ) ) - 4220 ) + 36697 15120 + ( - 137 60 + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ζ ( - 5 , a ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["-", "5"]], Zeta, Rule[Editable, True]], ",", TagBox["a", Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]]] + ζ ( - 5 ) + 1 252 ( 21 ( a - 1 ) 2 ( 2 ( a - 1 ) a - 1 ) a 2 + 1 ) TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + k = 0 1 - Re ( a ) ( ( a + k ) 5 log ( a + k ) - 1 2 ( ( a + k ) 2 ) 5 / 2 log ( ( a + k ) 2 ) ) 1 0 Zeta -5 a 120 PolyGamma -6 a -1 a -1 a -1 5 -1 1 2 2 a -1 5 -1 5 Glaisher a -1 4 10 D Zeta -3 -3 a -1 2 -1 5 2 a -1 2 2 Zeta 3 -1 3 Zeta 5 a -1 4 4 -1 42 a a 8275 -1 822 a a -5 a 10 -4220 36697 15120 -1 -1 137 60 Zeta -5 a D Zeta -5 -5 1 252 21 a -1 2 2 a -1 a -1 a 2 1 k 0 1 -1 a a k 5 a k -1 1 2 a k 2 5 2 log a k 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "5"]], ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["120", " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "6"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "a"]]]], "]"]]]], "+", FractionBox[RowBox[List["36697", "+", RowBox[List["42", " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "4220"]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List["8275", "-", RowBox[List["822", " ", "a", " ", RowBox[List["(", RowBox[List["10", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "+", "a"]], ")"]], " ", "a"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], "15120"], "+", RowBox[List[FractionBox["1", "252"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["21", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "2"], " ", SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", "a"]]]], ")"]]]]]], ")"]], " ", "EulerGamma"]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "5"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], "]"]]]], "-", RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "4"], " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "5"], " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", FractionBox[RowBox[List["5", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "2"], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Zeta", "[", "3", "]"]]]], "-", RowBox[List["3", " ", RowBox[List["Zeta", "[", "5", "]"]]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["\[Pi]", "4"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox["137", "60"]]], "+", "EulerGamma"]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "5"]], ",", "a"]], "]"]]]], "+", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "5"]], "]"]], "+", RowBox[List["10", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], "2"], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "3"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["1", "-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "5"], " ", RowBox[List["Log", "[", RowBox[List["a", "+", "k"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], ")"]], RowBox[List["5", "/", "2"]]], " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], "2"], "]"]]]]]], ")"]]]]]]]]]]

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

 2007-05-02