Quotient[k m, n] == k Quotient[m, n] + Sum[j (UnitStep[Mod[m/n, 1] - j/k] (1 - UnitStep[Mod[m/n, 1] - (j + 1)/k])), {j, 0, k - 1}] /; Element[k, Integers] && k >= 0 && Element[m/n, Reals]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Quotient", "[", RowBox[List[RowBox[List["k", " ", "m"]], ",", "n"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["k", " ", RowBox[List["Quotient", "[", RowBox[List["m", ",", "n"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List["j", " ", RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List[FractionBox["m", "n"], ",", "1"]], "]"]], "-", FractionBox["j", "k"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List[FractionBox["m", "n"], ",", "1"]], "]"]], "-", FractionBox[RowBox[List["j", "+", "1"]], "k"]]], "]"]]]], ")"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["k", ",", "Integers"]], "]"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List[FractionBox["m", "n"], ",", "Reals"]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mi> quotient </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> j </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <semantics> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <ci> $CellContext`m </ci> <apply> <power /> <ci> $CellContext`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> - </mo> <mfrac> <mi> j </mi> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <semantics> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> &#8290; </mo> <mi> mod </mi> <mo> &#8290; </mo> <mn> 1 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <ci> $CellContext`m </ci> <apply> <power /> <ci> $CellContext`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </annotation-xml> </semantics> <mo> - </mo> <mfrac> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> quotient </ci> <apply> <times /> <ci> k </ci> <ci> m </ci> </apply> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <ci> k </ci> <apply> <ci> quotient </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <ci> j </ci> <apply> <times /> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <rem /> <apply> <times /> <ci> $CellContext`m </ci> <apply> <power /> <ci> $CellContext`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> j </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <rem /> <apply> <times /> <ci> $CellContext`m </ci> <apply> <power /> <ci> $CellContext`n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> k </ci> <integers /> </apply> <apply> <in /> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Quotient", "[", RowBox[List[RowBox[List["k_", " ", "m_"]], ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["k", " ", RowBox[List["Quotient", "[", RowBox[List["m", ",", "n"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List["j", " ", RowBox[List["(", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List[FractionBox["m", "n"], ",", "1"]], "]"]], "-", FractionBox["j", "k"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List[FractionBox["m", "n"], ",", "1"]], "]"]], "-", FractionBox[RowBox[List["j", "+", "1"]], "k"]]], "]"]]]], ")"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List[FractionBox["m", "n"], "\[Element]", "Reals"]]]]]]]]]]

2001-10-29