|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/10.06.23.0001.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Sum[LerchPhiClassical[z^q, s, (a + k - 1)/q] z^(k - 1), {k, 1, q}] ==
q^s LerchPhiClassical[z, s, a] /; Element[q, Integers] && q > 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "q"], RowBox[List[RowBox[List["LerchPhiClassical", "[", RowBox[List[SuperscriptBox["z", "q"], ",", "s", ",", FractionBox[RowBox[List["a", "+", "k", "-", "1"]], "q"]]], "]"]], SuperscriptBox["z", RowBox[List["k", "-", "1"]]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["q", "s"], RowBox[List["LerchPhiClassical", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["q", ",", "Integers"]], "]"]], "\[And]", RowBox[List["q", ">", "0"]]]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <semantics> <mrow> <mover> <mi> Φ </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mi> q </mi> </msup> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[CapitalPhi]", "^"], "(", RowBox[List[TagBox[SuperscriptBox["z", "q"], Rule[Editable, True]], ",", TagBox["s", Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "k", "-", "1"]], "q"], Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> q </mi> <mi> s </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mover> <mi> Φ </mi> <mo> ^ </mo> </mover> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> s </mi> <mo> , </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[OverscriptBox["\[CapitalPhi]", "^"], "(", RowBox[List[TagBox["z", Rule[Editable, True]], ",", TagBox["s", Rule[Editable, True]], ",", TagBox["a", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2, $CellContext`e3], LerchPhi[$CellContext`e1, $CellContext`e2, $CellContext`e3]]]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> q </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <times /> <apply> <ci> LerchPhi </ci> <apply> <power /> <ci> z </ci> <ci> q </ci> </apply> <ci> s </ci> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> q </ci> <ci> s </ci> </apply> <apply> <ci> LerchPhi </ci> <ci> z </ci> <ci> s </ci> <ci> a </ci> </apply> </apply> </apply> <apply> <in /> <ci> q </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "q_"], RowBox[List[RowBox[List["LerchPhiClassical", "[", RowBox[List[SuperscriptBox["z_", "q_"], ",", "s_", ",", FractionBox[RowBox[List["a_", "+", "k_", "-", "1"]], "q_"]]], "]"]], " ", SuperscriptBox["z_", RowBox[List["k_", "-", "1"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["q", "s"], " ", RowBox[List["LerchPhiClassical", "[", RowBox[List["z", ",", "s", ",", "a"]], "]"]]]], "/;", RowBox[List[RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "0"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|