Legendre function of the second kind : Series representations (formula 07.10.06.0001)

LegendreQ

$Mathematica Notation$

Hypergeometric Functions

LegendreQ[nu,z]

Series representations

Generalized power series

Expansions at z==0

http://functions.wolfram.com/07.10.06.0001.01

Input Form

LegendreQ[\[Nu], z] == Sum[((Pochhammer[1/2, m] Pochhammer[1, m])/(m! Pochhammer[3/2, m])) Sum[((Pochhammer[-\[Nu], j] Pochhammer[1 + \[Nu], j])/j!) Sum[(((-1)^j Pochhammer[j - \[Nu], k] Pochhammer[1 + j + \[Nu], k])/ ((j + k)! k! 2^(j + k))) z^(j + 2 m + 1), {k, 0, Infinity}], {j, 0, Infinity}], {m, 0, Infinity}] + ((Sin[Pi \[Nu]] PolyGamma[\[Nu] + 1])/(2 Pi^(3/2))) Sum[(((-1)^j 2^j Gamma[(j - \[Nu])/2] Gamma[(1/2) (1 + j + \[Nu])])/j!) z^j, {j, 0, Infinity}] + Sum[((Pochhammer[-\[Nu], j] Pochhammer[1 + \[Nu], j])/j!) Sum[(((-1)^j Pochhammer[j - \[Nu], k] Pochhammer[1 + j + \[Nu], k])/ ((j + k)! k! 2^(j + k))) PolyGamma[1 + j + k] z^j, {k, 0, Infinity}], {j, 0, Infinity}] /; Abs[z] < 1

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List["1", ",", "m"]], "]"]]]], RowBox[List[RowBox[List["m", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "m"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]], " "]], RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["j", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "j", "+", "\[Nu]"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], "!"]], RowBox[List["k", "!"]], SuperscriptBox["2", RowBox[List["j", "+", "k"]]]]]], SuperscriptBox["z", RowBox[List["j", "+", RowBox[List["2", "m"]], "+", "1"]]]]]]]]]]]]]]], "+", " ", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], RowBox[List["2", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["2", "j"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["j", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "j", "+", "\[Nu]"]], ")"]]]], "]"]]]], RowBox[List["j", "!"]]], SuperscriptBox["z", "j"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]], " "]], RowBox[List["j", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["j", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "j", "+", "\[Nu]"]], ",", "k"]], "]"]], " "]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], "!"]], RowBox[List["k", "!"]], SuperscriptBox["2", RowBox[List["j", "+", "k"]]]]]], RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "+", "k"]], "]"]], SuperscriptBox["z", "j"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mi> π </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mi> j </mi> </msup> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> j </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> j </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> j </mi> </msup> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> j </mi> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "m"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "1", ")"]], "m"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "m"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "j"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mi> j </mi> <mo> + </mo> <mi> k </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> j </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LegendreQ </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <sin /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> Γ </ci> <apply> <times /> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> j </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> j </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> j </ci> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='integer'> 1 </cn> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 3 <sep /> 2 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> j </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> j </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> j </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["1", ",", "m"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["j", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "j", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["j", "+", RowBox[List["2", " ", "m"]], "+", "1"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], "!"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["2", RowBox[List["j", "+", "k"]]]]]]]]]], RowBox[List["j", "!"]]]]]]], RowBox[List[RowBox[List["m", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "m"]], "]"]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["2", "j"], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["j", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "j", "+", "\[Nu]"]], ")"]]]], "]"]]]], ")"]], " ", SuperscriptBox["z", "j"]]], RowBox[List["j", "!"]]]]]]], RowBox[List["2", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["j", "-", "\[Nu]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "j", "+", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "j", "+", "k"]], "]"]], " ", SuperscriptBox["z", "j"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "+", "k"]], ")"]], "!"]], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["2", RowBox[List["j", "+", "k"]]]]]]]]]], RowBox[List["j", "!"]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]

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

2001-10-29

LegendreQ[nu,mu,2,z]
LegendreQ[nu,mu,3,z]