ZetaClassical[s, a] == 1/(a + n)^s + Sum[1/(k - n + (a + n))^s, {k, 0, n - 1}] + Zeta[s] + Sum[((Pochhammer[1 - j - s, j] Zeta[j + s])/j!) (a + n)^j, {j, 1, Infinity}] /; Element[n, Integers] && n >= 0

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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> <mrow> <mover> <mi> &#950; </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <semantics> <mi> s </mi> <annotation encoding='Mathematica'> TagBox[&quot;s&quot;, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mi> a </mi> <annotation encoding='Mathematica'> TagBox[&quot;a&quot;, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> s </mi> </msup> </mfrac> </mrow> <mo> + </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> s </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;s&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> j </mi> <mo> - </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;j&quot;, &quot;-&quot;, &quot;s&quot;]], &quot;)&quot;]], &quot;j&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> s </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;j&quot;, &quot;+&quot;, &quot;s&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <ci> OverHat </ci> <ci> &#950; </ci> </apply> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <ci> s </ci> </apply> <apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> <ci> a </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> </apply> <ci> s </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Zeta </ci> <ci> s </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> s </ci> </apply> </apply> <ci> j </ci> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> j </ci> <ci> s </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> n </ci> </apply> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29