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This paper presents new integral representations as \ well as special values of the Barnes function. Moreover, the Barnes function \ is expressed in a closed form by means of the Hurwitz zeta function. These \ results can be used for numeric and symbolic computations of the Barnes \ function. " }], "Text", CellMargins->{{53.1875, 30.75}, {Inherited, Inherited}}, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["Preamble", "Section", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, PageBreakAbove->True, FontSize->16], Cell["\<\ In 1899, Barnes [1,2,3] introduced and studied the generalization of the \ Euler gamma function defined by the following functional equation:\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(TraditionalForm\`G( z + 1) = \(\[CapitalGamma](z)\)\ \(G( z)\), \ \ \ \ \ \(\(z\)\(\[Element]\)\(C\)\(\ \)\)\)], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[BoxData[ \(TraditionalForm\`\(\(G\ \((1)\)\)\(=\)\(1\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where \[CapitalGamma] is the gamma function. For the integer positive \ values of ", Cell[BoxData[ \(TraditionalForm\`z\)]], ", the Barnes G function is simply a product of factorials:" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(TraditionalForm\`G( n + 1) = \(\[Product]\+\(k = 1\)\%\(n - 1\)\(k!\)\ = \[Product]\+\(k \ = 1\)\%n \[CapitalGamma](k)\)\)], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ From the above functional equation and the Weierstrass canonical product for \ the gamma function, \ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(\[CapitalGamma](z)\), "=", RowBox[{\(1\/z\), " ", RowBox[{"exp", "(", RowBox[{\(-z\), TagBox["\[Gamma]", (EulerGamma&)]}], ")"}], " ", \(\[Product]\+\(m = 1\)\%\[Infinity]\ \(exp(z\/m)\)\/\(1 + \ z\/m\)\)}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["Barnes derived", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(TraditionalForm\`G( z + 1) = \((2\ \[Pi])\)\^\(z/2\)\ \(exp(\(-\(\(z + z\^2\ \((1 + \[Gamma])\)\)\/2\)\)\ )\)\ \(\[Product]\+\(k = \ 1\)\%\[Infinity]\((1 + z\/k)\)\^k\ \(exp(z\^2\/\(2\ k\) - z)\)\)\)], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where \[Gamma] is the Euler-Mascheroni constant. The above product is an \ entire function in a whole complex plane and can serve as an explicit \ definition of the Barnes function. Originally, the G function was introduced \ (in a different form) by Kinkelin [5] (see also Glaisher [6]) in his research \ on the asymptotics behavior of this product at ", Cell[BoxData[ \(TraditionalForm\`n \[Rule] \[Infinity]\)]], ":" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{\(1\^1\), SuperscriptBox["2", AdjustmentBox["2", BoxMargins->{{0, 0}, {0.4, -0.4}}, BoxBaselineShift->-0.4]]}], "..."}], " ", SuperscriptBox["n", AdjustmentBox["n", BoxMargins->{{0, 0}, {0.6, -0.6}}, BoxBaselineShift->-0.6]]}], "=", \(\(n!\)\^n\/\(G(n + 1)\)\)}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ Kinkelin also applied the theory of the G function to the class of \ trigonometric integrals \ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(\[Integral]\_0\%z\), RowBox[{"log", " ", RowBox[{ StyleBox["trig", FontSlant->"Italic"], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", StyleBox["trig(x)", FontSlant->"Italic"], " is any trigonometric or hyperbolic function. All such integrals can be \ expressed in finite terms of the G function. \n" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell["Alexeiewsky [7] generalized the Kinkelin product (1) to", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ RowBox[{ SuperscriptBox["1", SuperscriptBox["1", AdjustmentBox["p", BoxMargins->{{0, 0}, {0.581818, -0.581818}}, BoxBaselineShift->-0.581818]]], " ", SuperscriptBox["2", SuperscriptBox["2", AdjustmentBox["p", BoxMargins->{{0, 0}, {0.781818, -0.781818}}, BoxBaselineShift->-0.781818]]]}], "..."}], " ", SuperscriptBox[ StyleBox["n", FontSize->13], SuperscriptBox[ StyleBox["n", FontSize->9.5], AdjustmentBox["p", BoxMargins->{{0, 0}, {0.581818, -0.581818}}, BoxBaselineShift->-0.581818]]]}], "=", " ", RowBox[{"exp", StyleBox["(", FontSize->14], RowBox[{ TagBox[ RowBox[{" ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", RowBox[{ TagBox[\(-p\), (Editable -> True)], ",", TagBox[\(n + 1\), (Editable -> True)]}], ")"}]}], InterpretTemplate[ Zeta[ #, #2]&]], "-", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-p\), ")"}]}], " ", StyleBox[")", FontSize->14]}], " "}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["which is related to the multiple Barnes function.", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell["\<\ These days the G function remains the active topic of research (see [8--15].) \ The theory of the Barnes function has been related to certain spectral \ functions in mathematical physics, to the study of determinants of \ Laplacians, and to the Hecke L-functions. In [10] and [15] the G function is \ expressed in terms of the Hurwitz zeta function by\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(log\ \(G(z + 1)\) - z\ log\ \(\[CapitalGamma](z)\)\), "=", RowBox[{ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-1\), ")"}], "-", TagBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", RowBox[{ TagBox[\(-1\), (Editable -> True)], ",", TagBox["z", (Editable -> True)]}], ")"}], InterpretTemplate[ Zeta[ #, #2]&]]}]}], ",", " ", \(R(z)\ > \ 0\)}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ TagBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", RowBox[{"t", ",", TagBox["z", (Editable -> True)]}], ")"}], InterpretTemplate[ Zeta[ #, #2]&]], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ RowBox[{\(d\/\(d\ t\)\), TagBox[ RowBox[{"\[Zeta]", "(", RowBox[{"t", ",", TagBox["z", (Editable -> True)]}], ")"}], InterpretTemplate[ Zeta[ #, #2]&]]}], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-1\), ")"}], TraditionalForm]]], " is related to Glaisher's constant ", Cell[BoxData[ \(TraditionalForm\`A\)]], ":" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ A\), "=", RowBox[{ RowBox[{\(1\/12\), "-", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-1\), ")"}]}], "=", RowBox[{ FractionBox[ RowBox[{" ", RowBox[{ TagBox["\[Gamma]", (EulerGamma&)], " ", "+", \(log(2\ \[Pi])\)}]}], "12"], "-", FractionBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", "2", ")"}], \(2\ \[Pi]\^2\)]}]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ In [15] Adamchik obtained the closed form representation for the class of \ integrals involving cyclotomic polynomials and nested logarithms in terms of \ the Hurwitz zeta function. In the view of (3) the results can be translated \ into the G function notation, for example\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_0\%1\), RowBox[{\(\(\((x\^4 - 6\ x\^2 + 1)\)\(\ \)\)\/\((x\^2 + 1)\)\^3\), "log", " ", \(log(1\/x)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], "=", \(4\ \(log(\(G(3\/4)\)\/\(G(1\/4)\))\) - 3\ log\ \(\[CapitalGamma](1\/4)\) + log\ \(\[CapitalGamma](3\/4)\)\)}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "Choi ", StyleBox["et al.", FontSlant->"Italic"], " [13, 14] and Adamchik [15-17] considered a class of sums involving the \ Riemann zeta function which can be evaluated by means of the G function. Here \ is a series representation for the G function for ", Cell[BoxData[ \(TraditionalForm\`\[LeftBracketingBar]z\[RightBracketingBar] < 1\)]], "," }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(2\ log\ \(G(z + 1)\)\), "=", RowBox[{\(z\ \(log(2\ \[Pi])\)\), " ", "-", RowBox[{ TagBox["\[Gamma]", (EulerGamma&)], " ", \(z\^2\)}], "-", \(z\ \((z + 1)\)\), "+", RowBox[{"2", " ", RowBox[{ StyleBox[\(\[Sum]\+\(k = 2\)\%\[Infinity]\), FontSize->10], RowBox[{\(\((\(-1\))\)\^k\), " ", TagBox[\(\[Zeta](k)\), InterpretTemplate[ Zeta[ #]&]], \(\(\(\ \)\(z\^\(k + 1\)\)\)\/\(k + 1\)\)}]}]}]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["and for Glaisher's constant", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(log\ A\), "=", RowBox[{\(\(log\ 2\)\/12\), "+", " ", RowBox[{ StyleBox[\(\[Sum]\+\(\(\ \)\(k = 1\)\)\%\[Infinity]\), FontSize->14], RowBox[{ RowBox[{"(", FractionBox[ RowBox[{ TagBox[ RowBox[{"\[Zeta]", "(", TagBox[\(2\ k + 1\), (Editable -> True)], ")"}], InterpretTemplate[ Zeta[ #]&]], "-", "1"}], "36"], ")"}], \(\(\((4\ k + 7)\)\ \((7\ k + 8)\)\)\/\(\((k + 1)\)\ \((k + 2)\)\)\)}]}]}]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "Barnes [4] (followed by Vigneras [8] and Vardi [10]) generalized the G \ function to the multiple gamma function ", Cell[BoxData[ \(TraditionalForm\`G\_n\)]], " by the recurrence formula" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, PageBreakAbove->True, FontSize->12], Cell[BoxData[{ \(TraditionalForm\`\ \(G\_\(n + 1\)\)( z + 1) = \(\(G\_\(n + 1\)\)(z)\)\/\(\(G\_n\)(z)\), \ \ \ \ \ z \ \[Element] C\ , \ n \[Element] \[DoubleStruckCapitalN]\), "\[IndentingNewLine]", \(TraditionalForm\`\ \(\(\(G\_2\)(z)\)\(=\)\(G( z)\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\), "\[IndentingNewLine]", \(TraditionalForm\`\(\(\(G\_1\)( z)\)\(=\)\(1\/\(\(\[CapitalGamma]( z)\)\(\ \ \)\)\)\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)\)\)}], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "Here ", Cell[BoxData[ \(TraditionalForm\`G(z)\)]], " is the Barnes function and ", Cell[BoxData[ \(TraditionalForm\`\[CapitalGamma](z)\)]], " is the Euler gamma function. Vigneras and Vardi considered a slightly \ different form of the multiple gamma function defined as a reciprocal to the \ Barnes function ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalGamma]\_n\)(z)\ = \ 1\/\(\(G\_n\)(z)\)\)]], ". Vardi derived an implicit representation for ", Cell[BoxData[ \(TraditionalForm\`\(\[CapitalGamma]\_n\)(z)\)]], " in terms of the multiple zeta function. ", "In this paper we derive a closed form of ", Cell[BoxData[ \(TraditionalForm\`G\_n\)]], " in finite terms of the Hurwitz function and special polynomials. ", "Here are two particular cases rewritten in terms of the derivatives of the \ Hurwitz function when ", Cell[BoxData[ \(TraditionalForm\`n = 3\)]], " " }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(2\ log\ \(\(G\_3\)(z + 1)\) - \ z\^2\ log\ \(\[CapitalGamma](z)\)\), "=", RowBox[{\(\((1 - 2 z)\)\ \ log\ \(\(G\_2\)(z + 1)\)\), "+", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-2\), ")"}], "-", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(\(-2\), z\), ")"}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "and ", Cell[BoxData[ \(TraditionalForm\`\(\(n\)\(=\)\(4\)\(\ \)\)\)]] }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(6\ log\ \(\(G\_4\)(z + 1)\) - \(z\^3\) log\ \(\[CapitalGamma](z)\)\), " ", "=", " ", RowBox[{\(6\ \((1 - z)\)\ log\ \(\(G\_3\)(z + 1)\)\), "-", " ", "\[IndentingNewLine]", AdjustmentBox[" ", BoxMargins->{{0, 0}, {-0.299065, 0.299065}}, BoxBaselineShift->0.299065], AdjustmentBox[\(\((3\ z\^2 - 3\ z + 1)\)\ \ log\ \(\(G\_2\)( z + 1)\)\), BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], AdjustmentBox[" ", BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], AdjustmentBox["+", BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], AdjustmentBox[" ", BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], AdjustmentBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-3\), ")"}], BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], AdjustmentBox["-", BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], AdjustmentBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(\(-3\), z\), ")"}], BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], AdjustmentBox[" ", BoxMargins->{{0, 0}, {-1.64486, 1.64486}}, BoxBaselineShift->1.64486], "\[IndentingNewLine]"}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ We also present various integral and series representations as well as some \ special values of the Barnes function. These results can be used in numeric \ and symbolic computations of the G function. I believe the multiple gamma \ function is of direct interest for computer algebra researchers and users, \ because of its significant applications in physics, number theory, \ combinatorics and applied mathematics. The Barnes function deserves to be \ implemented in computer algebra systems. \ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["1. G function of the rational argument", "Section", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, FontSize->16], Cell["\<\ There are a few known special cases when the G function is expressible in a \ closed form. The first one is due to Barnes [1]:\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(TraditionalForm\`log\ \(G( 1\/2)\) = \(log\ 2\)\/24 - \(log\ \[Pi]\)\/4 - \(3\ log\ A\)\/2 + 1\/8\)], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["the other two are due to Choi and Srivastava [14]:", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[{ FormBox[ RowBox[{\(log\ \(G \((1\/4)\)\)\), "=", RowBox[{ RowBox[{"-", FractionBox[ TagBox["G", (Catalan&)], \(4\ \[Pi]\)]}], "-", \(3\/4\ log\ \(\[CapitalGamma](1\/4)\)\), "-", \(9\/8\ \ log\ A\), "+", \(3\/32\)}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{\(log\ \(G \((3\/4)\)\)\), "=", RowBox[{\(log\ \(G(5\/4)\)\), "+", FractionBox[ TagBox["G", (Catalan&)], \(2\ \[Pi]\)], "-", \(\(\(\ \)\(log\ \((2\ \[Pi]\^2)\)\)\)\/8\), " "}]}], TraditionalForm]}], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`G\)]], " is Catalan's constant. In a view of closed form representation for the \ derivatives of the Hurwitz functions, obtained in [18], and the formula (3), \ and it is easy to derive the additional special cases of ", Cell[BoxData[ \(TraditionalForm\`log\ \(G(p)\)\)]], " for ", Cell[BoxData[ FormBox[ StyleBox[\(p = 1\/3, \ 1\/6, \ 2\/3, \ 5\/6\), FontWeight->"Plain"], TraditionalForm]], FontWeight->"Bold"], "." }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[StyleBox["Proposition 1. ", FontWeight->"Bold"]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ \(G(1\/3)\)\), "=", RowBox[{\(\(log\ 3\)\/72\), "+", \(\[Pi]\/\(18\ \@3\)\), "-", \(2\/3\ log\ \(\[CapitalGamma](1\/3)\)\), "-", \(\(4\/3\) log\ A\), "-", FractionBox[ RowBox[{ SuperscriptBox[ TagBox["\[Psi]", PolyGamma], \((1)\)], "(", \(1\/3\), ")"}], \(12\ \@3\ \[Pi]\)], "+", \(1\/9\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ \(G(2\/3)\)\), "=", RowBox[{\(\(log\ 3\)\/72\), "+", \(\[Pi]\/\(18\ \@3\)\), "-", \(1\/3\ log\ \(\[CapitalGamma](2\/3)\)\), "-", \(\(4\/3\) log\ A\), "-", FractionBox[ RowBox[{ SuperscriptBox[ TagBox["\[Psi]", PolyGamma], \((1)\)], "(", \(2\/3\), ")"}], \(12\ \@3\ \[Pi]\)], "+", \(1\/9\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ \(G(1\/6)\)\), "=", RowBox[{\(-\(\(log\ 12\)\/144\)\), "+", \(\[Pi]\/\(20\ \@3\)\), "-", \(5\/6\ log\ \(\[CapitalGamma](1\/6)\)\), "-", \(\(5\/6\) log\ A\), "-", FractionBox[ RowBox[{ SuperscriptBox[ TagBox["\[Psi]", PolyGamma], \((1)\)], "(", \(1\/6\), ")"}], \(40\ \@3\ \[Pi]\)], "+", \(5\/72\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ \(G(5\/6)\)\), "=", RowBox[{\(-\(\(log\ 12\)\/144\)\), "+", \(\[Pi]\/\(20\ \@3\)\), "-", \(1\/6\ log\ \(\[CapitalGamma](5\/6)\)\), "-", \(\(5\/6\) log\ A\), "-", FractionBox[ RowBox[{ SuperscriptBox[ TagBox["\[Psi]", PolyGamma], \((1)\)], "(", \(5\/6\), ")"}], \(40\ \@3\ \[Pi]\)], "+", \(5\/72\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ StyleBox["where", FontSlant->"Italic"], " ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ SuperscriptBox[ TagBox["\[Psi]", PolyGamma], \((1)\)], "(", "z", ")"}], "=", RowBox[{ StyleBox[ FractionBox[\(\[PartialD]\^2\), \(\[PartialD]z\^2\), MultilineFunction->None], FontSize->13], " ", "log", " ", \(\[CapitalGamma](z)\)}]}], TraditionalForm]]], " ", StyleBox["is the polygamma function. ", FontSlant->"Italic"] }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell["\<\ Similar representations can be derived for the multiple G function.\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["2. Connection to the polylogarithm", "Section", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, FontSize->16], Cell["\<\ From the Lerch functional equation for the Hurwitz zeta function and formula \ (3), we can easily derive\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(log\ \((\(G(1 + z)\)\/\(G(1 - z)\))\)\), "=", RowBox[{ RowBox[{\(-\(\(\(\[Pi]\)\(\ \)\(i\)\(\ \)\)\/2\)\), " ", RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], "2"], "(", "z", ")"}]}], "+", \(z\ log\ \((\[Pi]\/\(sin(\[Pi]\ z)\))\)\), "+", \(\(\(\(i\)\(\ \)\)\/\(2\ \[Pi]\)\) \(\(Li\_2\)( e\^\(2\ \[Pi]\ i\ z\))\)\)}]}], ",", " ", \(0 < z < 1\)}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], "2"], "(", "z", ")"}], TraditionalForm]]], " is the second Bernoulli polynomial, and ", Cell[BoxData[ \(TraditionalForm\`\(Li\_2\)(z)\)]], " is the polylogarithm.The identity can be rewritten in an alternative form \ by means of the Clausen function ", Cell[BoxData[ \(TraditionalForm\`\(Cl\_2\)(z)\)]], ":" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(TraditionalForm\`log\ \((\(G(1 + z)\)\/\(G(1 - z)\))\) = z\ log\ \((\[Pi]\/\(sin(\[Pi]\ z)\))\) - \(1\/\(2\ \[Pi]\)\) \(\(Cl\_\(\ \(\ \)\(2\)\)\)(2\ \[Pi]\ z)\), \ \ 0 < z < 1\)], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["where", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(TraditionalForm\`\(Cl\_\(\(\ \)\(2\)\)\)( x) = \(-\(Im(\(Li\_2\)(e\^\(\(-i\)\ x\)))\)\)\)], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ Obviously enough, the G function is related to the Dirichlet L-series. Here \ is one of the formulas:\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(log\ \((\(G(7\/6)\)\/\(G(5\/6)\))\)\), "=", RowBox[{\(\(1\/6\) \(log(2\ \[Pi])\)\), "-", RowBox[{\(\(3\ \@3\)\/\(8 \[Pi]\)\), " ", RowBox[{ SubscriptBox["L", AdjustmentBox[\(-3\), BoxMargins->{{0, 0}, {-0.4, 0.4}}, BoxBaselineShift->0.4]], "(", "2", ")"}]}]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["3. Binet-like representation", "Section", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, FontSize->16], Cell["The Binet formula for the gamma function is", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{\(log\ \(\[CapitalGamma](z)\)\), "=", RowBox[{\(\((z - 1\/2)\)\ log\ z\), " ", "-", "z", "+", \(log(\@\(2\ \[Pi]\))\), " ", "+", RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{\(e\^\(\(-z\)\ x\)\/x\), " ", \((1\/\(1 - e\^\(-x\)\) - 1\/x - 1\/2)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "In this section we derive a similar representation for the Barnes \ function. Recall the well-known integral for ", Cell[BoxData[ FormBox[ TagBox[ RowBox[{"\[Zeta]", RowBox[{"(", RowBox[{ TagBox["s", (Editable -> True)], ",", TagBox["z", (Editable -> True)]}], ")"}]}], InterpretTemplate[ Zeta[ #, #2]&]], TraditionalForm]]], ":" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ TagBox[ RowBox[{"\[Zeta]", "(", RowBox[{ TagBox["s", (Editable -> True)], ",", TagBox["z", (Editable -> True)]}], ")"}], InterpretTemplate[ Zeta[ #, #2]&]], "=", \(\(1\/\(\[CapitalGamma]( s)\)\) \(\[Integral]\_0\%\[Infinity]\(\( x\^\(\(\ \)\(s - 1\ \)\)\ e\^\(\(-z\)\ x\)\)\/\(1 - e\^\(-x\)\)\) \[DifferentialD]x\)\)}], ",", " ", \(R(s) > 1\), ",", " ", \(R(z) > 0\)}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "The integral can be analytically continued to the domain ", Cell[BoxData[ \(TraditionalForm\`R(s) > \(-1\)\)]], ". To do so, we perform the standard procedure of removing the integrand \ singularity by subtracting the truncated Taylor series at ", Cell[BoxData[ \(TraditionalForm\`x = 0\)]], ":" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ TagBox[ RowBox[{"\[Zeta]", "(", RowBox[{ TagBox["s", (Editable -> True)], ",", TagBox["z", (Editable -> True)]}], ")"}], InterpretTemplate[ Zeta[ #, #2]&]], "=", RowBox[{\(-\(z\^\(1 - s\)\/\(1 - s\)\)\), "+", \(z\^\(-s\)\/2\), "+", \(s\/12\ z\^\(\(-s\) - 1\)\), "+", RowBox[{\(1\/\(\[CapitalGamma](s)\)\), RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{\(x\^\(s - 1\)\), " ", \(e\^\(\(-z\)\ x\)\), " ", \((1\/\(1 - e\^\(-x\)\) - 1\/x - x\/12 - 1\/2)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "Differentiating the above formula with respect to s and computing the \ limit at ", Cell[BoxData[ \(TraditionalForm\`s = \(-1\)\)]], ", we get " }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ TagBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", RowBox[{ TagBox[\(-1\), (Editable -> True)], ",", "z"}], ")"}], InterpretTemplate[ Zeta[ #, #2]&]], "=", RowBox[{\(1\/12\), "-", \(z\^2\/4\), "+", RowBox[{\(\(log\ z\)\/2\), RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], "2"], "(", "z", ")"}]}], "+", RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{\(e\^\(\(-z\)\ x\)\/x\^2\), " ", \((1\/\(1 - e\^\(-x\)\) - 1\/t - t\/12 - 1\/2)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}]}], TraditionalForm]], "Text",\ CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], "2"], "(", "z", ")"}], TraditionalForm]]], " is the second Bernoulli polynomial. From here it immediately follows " }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ StyleBox["Proposition 2. ", FontWeight->"Bold"], StyleBox["The Barnes G function admits the Binet integral representation:", FontSlant->"Italic"] }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(log\ \(G(z + 1)\)\), "=", RowBox[{\(z\ \ log\ \(\[CapitalGamma](z)\)\), " ", "-", " ", \(log\ A\), "+", \(z\^2\/4\), "-", RowBox[{\(\(log\ z\)\/2\), RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], "2"], "(", "z", ")"}]}], "-", " ", "\[IndentingNewLine]", RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{\(e\^\(\(-z\)\ x\)\/x\^2\), " ", \((1\/\(1 - e\^\(-x\)\) - 1\/t - 1\/2 - t\/12)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}]}], ",", " ", \(R(z) > 0\)}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ In a similar way we derive the representation for Glaisher's constant:\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ A\), " ", "=", " ", RowBox[{\(\(1 + log(2\ \[Pi])\)\/12\), "-", RowBox[{\(1\/\(2\ \[Pi]\^2\)\), RowBox[{\(\[Integral]\_0\%\[Infinity]\), RowBox[{\(\(x\ log\ x\)\/\(e\^x - 1\)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]], " "}]}]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["4. The multiple G function", "Section", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, PageBreakAbove->True, TextAlignment->Left, TextJustification->0, FontSize->16], Cell["\<\ For the multiple G function defined by the functional equation (4), Vardi \ [10] obtained the following formula\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ \(\(G\_n\)(z)\)\), "=", RowBox[{ RowBox[{"-", RowBox[{ StyleBox[ UnderscriptBox[ StyleBox["lim", FontSlant->"Italic"], \(s \[Rule] 0\)], FontSize->13, FontSlant->"Italic"], " ", \((\[PartialD]\(\(\[Zeta]\_n\)(s, \ z)\)\/\[PartialD]s)\)}]}], "-", RowBox[{\(\[Sum]\+\(k = 1\)\%n\), RowBox[{\(\((\(-1\))\)\^k\), " ", RowBox[{"(", "\[NoBreak]", GridBox[{ {"z"}, {\(k - 1\)} }], "\[NoBreak]", ")"}], " ", StyleBox[ SubscriptBox["R", AdjustmentBox[\(n + 1 - k\), BoxMargins->{{0, 0}, {-0.195122, 0.195122}}, BoxBaselineShift->0.195122]], FontSize->13]}]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["where", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(R\_n\), "=", RowBox[{\(\(\(\[Sum]\)\(\ \)\)\+\(\(k\)\(=\)\(1\)\(\ \)\)\%\(\(n\)\(\ \ \)\)\), RowBox[{ StyleBox[ UnderscriptBox[ StyleBox["lim", FontSlant->"Italic"], \(s \[Rule] 0\)], FontSize->13, FontSlant->"Italic"], " ", \((\[PartialD]\(\(\[Zeta]\_k\)(s, \ 1)\)\/\[PartialD]s)\)}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "and ", Cell[BoxData[ \(TraditionalForm\`\(\[Zeta]\_n\)(s, z)\)]], " is the multiple zeta function:" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(\(\[Zeta]\_n\)(s, z)\), "=", RowBox[{\(\(\[Sum]\+\(k\_1 = 0\)\%\[Infinity] ... \)\ \ \(\[Sum]\+\(k\_n = 0\)\%\[Infinity] 1\/\((\(k\_1 + k\_2 + ... \) + k\_n + z)\ \)\^s\)\), "=", RowBox[{\(\[Sum]\+\(k = 0\)\%\[Infinity]\), RowBox[{\(1\/\((k + z)\)\^s\), TagBox[ RowBox[{"(", GridBox[{ { TagBox[\(k + n - 1\), Identity, Editable->True]}, { TagBox[\(n - 1\), Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False]}]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "The aim of this section is to find a closed form representation for ", Cell[BoxData[ \(TraditionalForm\`\(G\_n\)(z)\)]], " in terms of the Hurwitz zeta function. For clarity of exposition, we \ first consider polynomials" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(\(P\_\(k, \ n\)\)(z)\), "=", RowBox[{\(\[Sum]\+\(i = k + 1\)\%n\), RowBox[{\(\((\(-z\))\)\^\(i - k - 1\)\), " ", RowBox[{ TagBox[ RowBox[{"(", GridBox[{ { TagBox[\(i - 1\), Identity, Editable->True]}, { TagBox["k", Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False], " ", StyleBox["[", FontSize->13], GridBox[{ {"n"}, {"i"} }], StyleBox["]", FontSize->13]}], " "}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["which can be rewritten in the alternative form", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(\(P\_\(k, \ n\)\)(z)\), "=", RowBox[{\(\[Sum]\+\(i = 1\)\%n\), RowBox[{ TagBox[ RowBox[{"(", GridBox[{ { TagBox["z", Identity, Editable->True]}, { TagBox[\(n - i\), Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False], RowBox[{\(\(\(\ \)\(\((n - 1)\)!\)\(\ \ \)\)\/\(\((i - 1)\)!\)\), StyleBox["[", FontSize->13], GridBox[{ {"i"}, {\(k + 1\)} }], StyleBox["]", FontSize->13]}], StyleBox[" ", FontSize->13]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{"[", GridBox[{ { AdjustmentBox["n", BoxMargins->{{0, 0}, {-0.285714, 0.285714}}, BoxBaselineShift->0.285714]}, { AdjustmentBox["i", BoxMargins->{{0, 0}, {0.285714, -0.285714}}, BoxBaselineShift->-0.285714]} }], "]"}], TraditionalForm]]], " are unsigned Stirling numbers of the first kind. The polynomials ", Cell[BoxData[ \(TraditionalForm\`\(P\_\(k, \ n\)\)(z)\)]], " satisfy the functional equation" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ \(TraditionalForm\`\(P\_\(k, \ n + 1\)\)(z) - n\ \(\(P\_\(k, \ n\)\)(z)\) - \(P\_\(k, \ n + 1\)\)(z + 1)\ = 0\)], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ AdjustmentBox[\(P\_\(k, \ n\)\), BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], "(", AdjustmentBox["z", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], AdjustmentBox[")", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857]}], AdjustmentBox["=", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], AdjustmentBox["0", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857]}], AdjustmentBox[",", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], AdjustmentBox[" ", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], RowBox[{ AdjustmentBox["k", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], AdjustmentBox["\[GreaterEqual]", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], AdjustmentBox["n", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], StyleBox[ RowBox[{ AdjustmentBox[" ", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857], AdjustmentBox[" ", BoxMargins->{{0, 0}, {-0.428571, 0.428571}}, BoxBaselineShift->0.428571], " "}]]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\ \(\[IndentingNewLine]\)\(\[IndentingNewLine]\)\)\)], "Input", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}], Cell["Here are the main properties of these polynomials:", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[{ FormBox[ RowBox[{ RowBox[{\(\(P\_\(n - 1, \ n\)\)(z)\), "=", RowBox[{\(1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(\(P\_\(k, \ n\)\)( 1)\)\), "=", RowBox[{ StyleBox["[", FontSize->13], GridBox[{ {\(n - 1\)}, {"k"} }], StyleBox["]", FontSize->13]}]}]}], StyleBox[ RowBox[{ StyleBox[" ", FontSize->13], " "}]]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{\(\(P\_\(n - 2, \ n\)\)(z)\), "=", RowBox[{ RowBox[{\(z(1 - n)\), " ", "+", " ", RowBox[{\(\(n(n - 1)\)\/2\), StyleBox[" ", FontWeight->"Bold"], \(\(P\_\(0, \ n\)\)(z)\)}]}], "=", " ", RowBox[{ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(n - z - 1\)}, {\(n - 1\)} }], "\[NegativeThinSpace]", ")"}], \(\((n - 1)\)!\)}]}]}], " "}], TraditionalForm], "\[IndentingNewLine]", FormBox[\(\(P\_\(0, \ n\)\)(1\/2) = \(\((2\ n - 3)\)!!\)\/2\^\(n - 1\)\), TraditionalForm]}], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, FontSize->12], Cell[TextData[{ StyleBox["\nLemma 1.", FontWeight->"Bold"], " ", StyleBox["The multiple zeta function ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`\(\[Zeta]\_n\)(s, z)\)], FontSlant->"Italic"], StyleBox[" defined by (19) may be expressed by means of the Hurwitz \ function", FontSlant->"Italic"] }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(\(\[Zeta]\_n\)(s, z)\), "=", RowBox[{\(1\/\(\((n - 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1\) = \((\((k + z)\) - z)\)\^\(i - 1\)\)]], " by the binomial theorem, implies that" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ TagBox[ RowBox[{"(", GridBox[{ { TagBox[\(k + n - 1\), Identity, Editable->True]}, { TagBox[\(n - 1\), Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False], "=", RowBox[{\(1\/\(\((n - 1)\)!\)\), RowBox[{ AdjustmentBox[ UnderoverscriptBox[ StyleBox["\[Sum]", FontSize->9], \(i = 0\), "n"], BoxMargins->{{-0.0769231, 0.0769231}, {0.285714, -0.285714}}, BoxBaselineShift->-0.285714], RowBox[{ UnderoverscriptBox[ StyleBox["\[Sum]", FontSize->13], \(j = 0\), \(i - 1\)], RowBox[{ TagBox[ RowBox[{"(", GridBox[{ { TagBox[\(i - 1\), Identity, Editable->True]}, { TagBox["j", Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False], " ", \(\((k + z)\)\^j\), " ", RowBox[{\(\((\(-z\))\)\^\(i - j - 1\)\), " ", StyleBox["[", FontSize->13], GridBox[{ {"n"}, {"i"} }], StyleBox["]", FontSize->13]}]}]}]}]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ Interchanging the order of summation and making use of (20), we obtain\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ TagBox[ RowBox[{"(", GridBox[{ { TagBox[\(k + n - 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1)\)!\)\), RowBox[{\(\[Sum]\+\(j = 0\)\%n\), RowBox[{ TagBox[ RowBox[{ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", TagBox[\(-j\), (Editable -> True)], ")"}], " "}], InterpretTemplate[ Zeta[ #, #2]&]], StyleBox["[", FontSize->13], GridBox[{ {\(n - 1\)}, {"j"} }], StyleBox["]", FontSize->13]}], StyleBox[" ", FontSize->13]}]}]}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["Further, in a view of (18), we have", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(R\_n\), "=", RowBox[{ RowBox[{\(\(\(\[Sum]\)\(\ \)\)\+\(\(k\)\(=\)\(1\)\(\ \ \)\)\%\(\(n\)\(\ \)\)\), \(1\/\(\((k - 1)\)!\)\), RowBox[{\(\[Sum]\+\(j = 0\)\%k\), " ", RowBox[{ TagBox[ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", TagBox[\(-j\), (Editable -> True)], ")"}], InterpretTemplate[ Zeta[ #, #2]&]], " ", StyleBox["[", FontSize->13], GridBox[{ {\(k - 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Integral representations via polygammas", "Section", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, FontSize->16], Cell["\<\ In [15] Adamchik derived a closed form solution to the integral\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_0\%z\), RowBox[{\(x\^n\), " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], ",", " ", \(R(z)\ > \ 0\), ",", " ", \(n\ \[Element] \[DoubleStruckCapitalN]\)}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["in finite terms of the Hurwitz zeta function:", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_0\%z\), RowBox[{\(x\^n\), " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], "=", RowBox[{ RowBox[{\(\((\(-1\))\)\^\(n - 1\)\), " ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-n\), ")"}]}], "+", \(\(\(\(\((\(-1\))\)\^n\)\(\ \)\)\/\(n + 1\)\) B\_\(n + 1\)\ H\_n\), "+", " ", "\[IndentingNewLine]", " ", RowBox[{\(\[Sum]\+\(k = 0\)\%n\), RowBox[{\(\((\(-1\))\)\^k\), " ", TagBox[ RowBox[{"(", GridBox[{ { TagBox["n", Identity, Editable->True]}, { TagBox["k", Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False], " ", \(z\^\(n - k\)\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(\(-k\), z\), ")"}], "-", FractionBox[ RowBox[{" ", RowBox[{ RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], \(k + 1\)], "(", "z", ")"}], " ", \(H\_k\), " "}]}], \(k + 1\)]}], ")"}]}]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`B\_n\)]], "and ", Cell[BoxData[ \(TraditionalForm\`H\_n\)]], " are Bernoulli and Harmonic numbers respectively. If ", Cell[BoxData[ \(TraditionalForm\`n = 1\)]], " the integral (26) leads to the following representation for the Barnes G \ function:" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(log\ \(G(z + 1)\)\), "=", RowBox[{\(\(z - z\^2\)\/2\), "+", \(z\ \(log(\@\(2\ \[Pi]\))\)\), "+", RowBox[{\(\[Integral]\_0\%z\), RowBox[{"x", " ", \(\[Psi](x)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}]}], ",", " ", \(R(z)\ > \ \(-1\)\)}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "This representation demonstrates that the complexity of computing ", Cell[BoxData[ \(TraditionalForm\`G(z)\)]], " depends at most on the complexity of computing the polygamma function. \ The restriction ", Cell[BoxData[ \(TraditionalForm\`R(z) > \(-1\)\)]], " can be easily removed by analyticity of the polygamma. For example, by \ resolving the singularity of the integrand at the pole ", Cell[BoxData[ \(TraditionalForm\`x = \(-1\)\)]], ", we continue ", Cell[BoxData[ \(TraditionalForm\`log\ \(G(z + 1)\)\)]], " to the wider area ", Cell[BoxData[ \(TraditionalForm\`R(z) > \(-2\)\)]], ":" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ \(G(z + 1)\)\), "=", RowBox[{\(\(z - z\^2\)\/2\), "+", \(z\ \(log(\@\(2\ \[Pi]\))\)\), "+", \(log\ \((z + 1)\)\), " ", "+", \(\[Integral]\_0\%z x\ \(\[Psi](x)\)\), "-", RowBox[{\(1\/\(x + 1\)\), StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["\<\ Another way to continue (27) into the left half-plane is to use the \ identity\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], "=", RowBox[{ RowBox[{ TagBox["\[Psi]", PolyGamma], "(", \(-x\), ")"}], "-", \(\[Pi]\ \(cot(\[Pi]\ x)\)\), "-", \(1\/x\)}]}], TraditionalForm]], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["which upon substituting it into (27) yields", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(log\ \(G(1 + z)\)\), "=", RowBox[{\(log\ \(G(1 - z)\)\), "+", \(z\ \(log(2\ \[Pi])\)\), "-", RowBox[{\(\[Integral]\_0\%z\), RowBox[{"\[Pi]", " ", "x", " ", "cot", " ", \((\[Pi]\ x)\), " ", StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "The identity (29) holds everywhere in a complex plane of ", Cell[BoxData[ \(TraditionalForm\`z\)]], ", except the real axes, where the integrand has simple poles. Therefore, \ in a view of (29) we can continue (27) to \[DoubleStruckCapitalC] / ", Cell[BoxData[ \(TraditionalForm\`\(\[DoubleStruckCapitalR]\^-\)\)]], ". Note, if ", Cell[BoxData[ \(TraditionalForm\`z\)]], " is negative we can still use the representation (27), but with a contour \ of integration deformed in such a way that it does not cross poles." }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ "For ", Cell[BoxData[ \(TraditionalForm\`n = 2\)]], ", the formula (26) yields" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_0\%z\), RowBox[{\(x\^2\), " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], "=", RowBox[{\(\(-2\)\ log\ \(\(G\_3\)(z + 1)\)\), " ", "+", \(log\ \(\(G\_2\)(z + 1)\)\), " ", "+", " ", "\[IndentingNewLine]", RowBox[{\(z\^2\), RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], "-", RowBox[{"2", " ", "z", " ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-1\), ")"}]}], "+", \(z\/12\ \((6\ z\^2 - 3\ z - 1)\)\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{54.1875, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "and for ", Cell[BoxData[ \(TraditionalForm\`n = 3\)]], " :" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_0\%z\), RowBox[{\(x\^3\), " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], "=", RowBox[{\(6\ log\ \(\(G\_4\)(z + 1)\)\), " ", "-", \(6\ log\ \(\(G\_3\)(z + 1)\)\), " ", "+", \(log\ \(\(G\_2\)(z + 1)\)\), " ", "+", "\[IndentingNewLine]", " ", RowBox[{\(z\^3\), RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], " ", "-", RowBox[{"3", " ", \(z\^2\), RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-1\), ")"}]}], "+", RowBox[{"3", "z", " ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-2\), ")"}]}], "+", \(z\^2\/24\ \ \((11\ z\^2 - 4\ z - 1)\)\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell["\<\ The general formula can also be derived. Skipping the technical details, we \ have\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ StyleBox["Proposition 4.", FontWeight->"Bold"], " ", StyleBox["Let ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`n\)], FontSlant->"Italic"], StyleBox[" be a positive integer and and ", FontSlant->"Italic"], Cell[BoxData[ \(TraditionalForm\`R(z) > \(-1\)\)], FontSlant->"Italic"], StyleBox[", then", FontSlant->"Italic"] }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{\(\[Integral]\_0\%z\), RowBox[{\(x\^n\), " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], "=", " ", RowBox[{ RowBox[{"-", RowBox[{\(\[Sum]\+\(k = 1\)\%n\), RowBox[{\(\((\(-1\))\)\^k\), " ", \(k!\), " ", RowBox[{ StyleBox["{", FontSize->14], GridBox[{ {"n"}, { AdjustmentBox["k", BoxMargins->{{0, 0}, {0.142857, -0.142857}}, BoxBaselineShift->-0.142857]} }], StyleBox["}", FontSize->14]}], " ", "log", " ", \(\(G\_\(k + 1\)\)(z + 1)\)}]}]}], "+", " ", "\[IndentingNewLine]", " ", RowBox[{ UnderoverscriptBox[ StyleBox["\[Sum]", FontSize->8], \(\(\ \)\(k = 0\)\), \(\(\ \ \ \)\(n - 1\)\)], RowBox[{\(\((\(-1\))\)\^k\), " ", TagBox[ RowBox[{"(", GridBox[{ { TagBox["n", Identity, Editable->True]}, { TagBox["k", Identity, Editable->True]} }], ")"}], InterpretTemplate[ Binomial[ #, #2]&], Editable->False], " ", \(z\^\(n - k\)\), " ", RowBox[{"(", RowBox[{ RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-k\), ")"}], " ", "-", FractionBox[ RowBox[{ RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], \(k + 1\)], "(", "z", ")"}], " ", \(H\_k\)}], \(k + 1\)]}], ")"}]}]}], "+", RowBox[{\(\((\(-1\))\)\^n\), " ", \(H\_n\), " ", FractionBox[ RowBox[{\(B\_\(n + 1\)\), "-", RowBox[{ SubscriptBox[ TagBox["B", BernoulliB], \(n + 1\)], "(", "z", ")"}]}], \(n + 1\)]}]}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ StyleBox["where ", FontSlant->"Italic"], Cell[BoxData[ FormBox[ RowBox[{"{", GridBox[{ { AdjustmentBox["n", BoxMargins->{{0, 0}, {-0.285714, 0.285714}}, BoxBaselineShift->0.285714]}, { AdjustmentBox["k", BoxMargins->{{0, 0}, {0.285714, -0.285714}}, BoxBaselineShift->-0.285714]} }], "}"}], TraditionalForm]], FontSlant->"Italic"], StyleBox[" are Stirling numbers of the second kind. ", FontSlant->"Italic"] }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ "We can resolve the equation (32) with respect to ", Cell[BoxData[ \(TraditionalForm\`\(G\_\(k + 1\)\)(z + 1)\)]], ". In particular, combining (27) with (30) leads us to the following \ integral representation for the triple Barnes function:" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(2\ log\ \(\(G\_3\)(z + 1)\)\), " ", "=", RowBox[{ RowBox[{\(\[Integral]\_0\%z\), RowBox[{\(x(1 - x)\), " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], "-", RowBox[{"z", " ", \((1 - z)\), RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], "-", " ", "\[IndentingNewLine]", RowBox[{"2", " ", "z", " ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-1\), ")"}]}], "+", \(\(z\/12\)\(\ \)\((6\ z\^2 - 9\ z + 5)\)\(\ \)\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell["and combining (27), (30) and (31) yields:", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(6\ \ log\ \(\(G\_4\)(z + 1)\)\), "=", RowBox[{ RowBox[{\(\[Integral]\_0\%z\), RowBox[{"x", " ", \((x - 1)\), " ", \((x - 2)\), " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}], "-", RowBox[{\(z(z - 1)\), " ", \((z - 2)\), " ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", "0", ")"}]}], " ", "+", " ", RowBox[{"3", "z", " ", \((z - 2)\), " ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-1\), ")"}]}], "-", RowBox[{"3", " ", "z", " ", RowBox[{ SuperscriptBox["\[Zeta]", "\[Prime]", MultilineFunction->None], "(", \(-2\), ")"}]}], "-", \(\(\(z\)\(\ \)\)\/24\ \((11\ z\^3 - 40\ z\^2 + 41\ z - 18)\)\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "Formulas (33) and (34) are suitable for numeric computation of ", Cell[BoxData[ \(TraditionalForm\`\(G\_3\)(z)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\(G\_4\)(z)\)]], " for any ", Cell[BoxData[ \(TraditionalForm\`z \[Element] \[DoubleStruckCapitalC]/\(\ \[DoubleStruckCapitalR]\^-\)\)]], "." }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12] }, Open ]], Cell[CellGroupData[{ Cell["6. Implementation remarks", "Section", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, FontSize->16], Cell[TextData[{ "We have presented here many available results on the Barnes function as \ well as new results on numeric and symbolic computations. In this section we \ discuss a numeric computational scheme for the Barnes G function. The \ integral representation via polygamma functions considered in the previous \ section seems to be an efficient numeric procedure for evaluating G \ function. Based on (27), we define the double Barnes function ", Cell[BoxData[ \(TraditionalForm\`G(z) = \(G\_2\)(z)\)]], " as" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{"\[IndentingNewLine]", RowBox[{ RowBox[{\(G(z)\), "=", RowBox[{ SuperscriptBox[\((2\ \[Pi])\), StyleBox[ AdjustmentBox[\(\(z - 1\)\/2\), BoxMargins->{{0, 0}, {1.23376, -1.23376}}, BoxBaselineShift->-1.23376], FontSize->13]], " ", RowBox[{"exp", "(", RowBox[{\(-\(\(\(\ \)\(\((z - 1)\) \((z - 2)\)\)\)\/2\)\), "+", RowBox[{\(\[Integral]\_0\%\(z - 1\)\), RowBox[{"x", " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}], ")"}]}]}], ",", " ", \(arg(z) \[NotEqual] \[Pi]\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "This representation is valid for ", Cell[BoxData[ \(TraditionalForm\`z \[Element] \[DoubleStruckCapitalC]/\(\ \[DoubleStruckCapitalR]\^-\)\)]], ". If ", Cell[BoxData[ \(TraditionalForm\`z\)]], " is a negative real, we have" }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[{ FormBox[ RowBox[{ AdjustmentBox[\(G(\(-n\)) = 0, \ \ \ \ \ \ \ n \[Element] \[DoubleStruckCapitalN]\), BoxMargins->{{0, 0}, {-1, 1}}, BoxBaselineShift->1], StyleBox[ RowBox[{ AdjustmentBox[" ", BoxMargins->{{0, 0}, {-1, 1}}, BoxBaselineShift->1], " \ "}]], "\[IndentingNewLine]", " \ "}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{ RowBox[{\(G(z)\), "=", RowBox[{ SuperscriptBox[\((2\ \[Pi])\), StyleBox[ AdjustmentBox[\(\(z - 1\)\/2\), BoxMargins->{{0, 0}, {1.23376, -1.23376}}, BoxBaselineShift->-1.23376], FontSize->13]], RowBox[{"exp", "(", RowBox[{\(-\(\(\(\ \)\(\((z - 1)\) \((z - 2)\)\)\)\/2\)\), "+", RowBox[{\(\[Integral]\_\[Gamma]\), RowBox[{"x", " ", RowBox[{ TagBox["\[Psi]", PolyGamma], "(", "x", ")"}], StyleBox[ RowBox[{"d", StyleBox["x", FontSlant->"Italic"]}]]}]}]}], ")"}]}]}], ",", " ", \(arg \((z)\) = \[Pi]\)}], TraditionalForm]}], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where the contour of integration \[Gamma] is a line between 0 and ", Cell[BoxData[ \(TraditionalForm\`z - 1\)]], " that does not cross the negative real axis; for example, \[Gamma] could \ be the following path: ", Cell[BoxData[ \(TraditionalForm\`{0, i, i + z - 1, z - 1}\)]], ". " }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell["", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell["\<\ Another method of defining the Barnes G function for negative reals is to use \ the formula (12). This gives\ \>", "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[BoxData[ FormBox[ RowBox[{\(G(\(-z\))\), "=", RowBox[{ SuperscriptBox[\((\(-1\))\), RowBox[{ RowBox[{ AdjustmentBox["\[LeftFloor]", BoxMargins->{{0, 0}, {0.207792, -0.207792}}, BoxBaselineShift->-0.207792], AdjustmentBox[\(z/2\), BoxMargins->{{0, 0}, {0.423377, -0.423377}}, BoxBaselineShift->-0.423377], AdjustmentBox["\[RightFloor]", BoxMargins->{{0, 0}, {0.207792, -0.207792}}, BoxBaselineShift->-0.207792]}], AdjustmentBox["-", BoxMargins->{{0, 0}, {0.207792, -0.207792}}, BoxBaselineShift->-0.207792], AdjustmentBox["1", BoxMargins->{{0, 0}, {0.207792, -0.207792}}, BoxBaselineShift->-0.207792]}]], " ", \(G(z + 2)\), " ", \(\((\[LeftBracketingBar]sin(\[Pi]\ z)\[RightBracketingBar]\/\ \[Pi])\)\^\(z + 1\)\), " ", \(exp(\(1\/\(2\ \[Pi]\)\) \(\(Cl\_2\)( 2\ \[Pi]\ \((z - \[LeftFloor]z\[RightFloor])\))\))\)}]}], TraditionalForm]], "NumberedEquation", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, TextAlignment->Center, TextJustification->0, FontSize->12], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\[LeftFloor]z\[RightFloor]\)]], " is the floor function, and ", Cell[BoxData[ \(TraditionalForm\`\(Cl\_2\)(z)\)]], " is the Clausen function. 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E. W. Barnes, The theory of the G-finction, ", StyleBox["Quart. J. Math.,", FontSlant->"Italic"], " ", StyleBox["31", FontWeight->"Bold"], "(1899), 264-314." }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ "2. E. W. Barnes, Genesis of the double gamma function, ", StyleBox["Proc. London Math. Soc.,", FontSlant->"Italic"], " ", StyleBox["31", FontWeight->"Bold"], "(1900), 358-381." }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ "3. E. W. Barnes, The theory of the double gamma function, ", StyleBox["Philos. Trans. Roy. Soc. London, Ser. A,", FontSlant->"Italic"], " ", StyleBox["196", FontWeight->"Bold"], "(1901), 265-388." }], "Text", CellMargins->{{Inherited, 30.75}, {Inherited, Inherited}}, FontSize->12], Cell[TextData[{ "4. E. W. Barnes, On he theory of the multiple gamma function, ", StyleBox["Trans. Cambridge Philos. 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