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 LerchPhi

 http://functions.wolfram.com/10.06.03.0058.01

 Input Form

 LerchPhi[z, n, a] == ((1 - (1 - (-1)^n) UnitStep[-Re[a]]) HypergeometricPFQ[{1, Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, n]}, {1 + Subscript[a, 1], 1 + Subscript[a, 2], \[Ellipsis], 1 + Subscript[a, n]}, z])/a^n + UnitStep[-Re[a]] (1 - (-1)^n) z^Floor[-Re[a]] ((z HypergeometricPFQ[{1, Subscript[b, 1], Subscript[b, 2], \[Ellipsis], Subscript[b, n]}, {1 + Subscript[b, 1], 1 + Subscript[b, 2], \[Ellipsis], 1 + Subscript[b, n]}, z])/(a + Floor[-Re[a]] + 1)^n + (UnitStep[Im[a]] (1 + Floor[-Re[a]] + Floor[Re[a]]))/ ((Floor[-Re[a]] + a)^2)^(n/2)) /; Subscript[b, n] == Subscript[a, n] + Floor[-Re[Subscript[a, n]]] + 1 && Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n] == a && Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 Φ ( z , n , a ) TagBox[RowBox[List["\[CapitalPhi]", "(", RowBox[List[TagBox["z", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["n", LerchPhi, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["a", LerchPhi, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$, ZetaDump`e3\$], LerchPhi[ZetaDump`e1\$, ZetaDump`e2\$, ZetaDump`e3\$]]], Rule[Editable, False], Rule[Selectable, False]] ( 1 - ( 1 - ( - 1 ) n ) θ UnitStep ( - Re ( a ) ) ) a - n n + 1 F n ( 1 , a 1 , a 2 , , a n ; a 1 + 1 , a 2 + 1 , , a n + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["n", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["n", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "n"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] + θ UnitStep ( - Re ( a ) ) ( 1 - ( - 1 ) n ) z - Re ( a ) ( z ( a + - Re ( a ) + 1 ) - n n + 1 F n ( 1 , b 1 , b 2 , , b n ; b 1 + 1 , b 2 + 1 , , b n + 1 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["n", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["n", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "n"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "2"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["b", "n"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] + ( - Re ( a ) + Re ( a ) + 1 ) θ UnitStep ( Im ( a ) ) ( ( a + - Re ( a ) ) 2 ) n / 2 ) /; b n - Re ( a n ) + a n + 1 a 1 a 2 a n a n + Condition LerchPhi z n a 1 -1 1 -1 -1 n UnitStep -1 a a -1 n HypergeometricPFQ 1 Subscript a 1 Subscript a 2 Subscript a n Subscript a 1 1 Subscript a 2 1 Subscript a n 1 z UnitStep -1 a 1 -1 -1 n z -1 a z a -1 a 1 -1 n HypergeometricPFQ 1 Subscript b 1 Subscript b 2 Subscript b n Subscript b 1 1 Subscript b 2 1 Subscript b n 1 z -1 a a 1 UnitStep a a -1 a 2 n 2 -1 -1 Subscript b n -1 Subscript a n Subscript a n 1 Subscript a 1 Subscript a 2 Subscript a n a n SuperPlus [/itex]

 Rule Form

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 Date Added to functions.wolfram.com (modification date)

 2007-05-02