Integrate[Subscript[OverTilde[B], m][t/a] Subscript[OverTilde[B], n][t/b], {t, 0, a b}] == ((-1)^(n - 1) m! n! GCD[a, b]^(m + n) BernoulliB[m + n])/ ((m + n)! a^(m - 1) b^(n - 1)) /; Subscript[OverTilde[B], n][x] == BernoulliB[n, Mod[x, 1]] && Element[a, Integers] && a > 1 && Element[b, Integers] && b > 1 && Element[m, Integers] && m > 1 && Element[n, Integers] && n > 1

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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> <msubsup> <mo> &#8747; </mo> <mn> 0 </mn> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </msubsup> <mrow> <mrow> <mrow> <msub> <mover> <mi> B </mi> <mo> ~ </mo> </mover> <mi> m </mi> </msub> <mo> ( </mo> <mfrac> <mi> t </mi> <mi> a </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mover> <mi> B </mi> <mo> ~ </mo> </mover> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <mi> t </mi> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> b </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mi> gcd </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mover> <mi> B </mi> <mo> ~ </mo> </mover> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <semantics> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <ci> a </ci> <ci> b </ci> </apply> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <apply> <ci> OverTilde </ci> <ci> B </ci> </apply> <ci> m </ci> </apply> <apply> <times /> <ci> t </ci> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <apply> <ci> OverTilde </ci> <ci> B </ci> </apply> <ci> n </ci> </apply> <apply> <times /> <ci> t </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <ci> a </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <gcd /> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <ci> BernoulliB </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <apply> <ci> OverTilde </ci> <ci> B </ci> </apply> <ci> n </ci> </apply> <ci> x </ci> </apply> <apply> <ci> BernoulliB </ci> <ci> n </ci> <apply> <rem /> <ci> $CellContext`x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <integers /> </apply> <apply> <in /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <integers /> </apply> <apply> <in /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29