PolyGamma[-7, 1] == (1/604800) (-211 + 5040 Log[Glaisher] + 420 Log[2 Pi] + (3150 Zeta[3])/Pi^2 - (9450 Zeta[5])/Pi^4 - 5040 Derivative[1][Zeta][-5] - 16800 Derivative[1][Zeta][-3])

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mrow> <mn> 5040 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <semantics> <mi> A </mi> <annotation encoding='Mathematica'> TagBox[&quot;A&quot;, Function[List[], Glaisher]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 420 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 3150 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mrow> <mn> 5040 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 16800 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> &#950; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mn> 9450 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 5 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;5&quot;, Zeta, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mfrac> <mo> - </mo> <mn> 211 </mn> </mrow> <mn> 604800 </mn> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyGamma </ci> <cn type='integer'> -7 </cn> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 5040 </cn> <apply> <ln /> <ci> Glaisher </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 420 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3150 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5040 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -5 </cn> </apply> <cn type='integer'> -5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16800 </cn> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <cn type='integer'> -3 </cn> </apply> <cn type='integer'> -3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9450 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -211 </cn> </apply> <apply> <power /> <cn type='integer'> 604800 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02