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 LegendreQ

 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

 Q TagBox["Q", LegendreQ] ν ( z ) sin ( π ν ) ψ TagBox["\[Psi]", PolyGamma] ( ν + 1 ) 2 π 3 / 2 j = 0 ( - 1 ) j 2 j j ! Γ ( j - ν 2 ) Γ ( j + ν + 1 2 ) z j + j = 0 ( - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] j ! k = 0 ( - 1 ) j ( j - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( j + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( j + k + 1 ) z j ( j + k ) ! k ! 2 j + k + m = 0 ( 1 2 ) m TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "m"], Pochhammer] ( 1 ) m TagBox[SubscriptBox[RowBox[List["(", "1", ")"]], "m"], Pochhammer] m ! ( 3 2 ) m TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "m"], Pochhammer] j = 0 ( - ν ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", "\[Nu]"]], ")"]], "j"], Pochhammer] ( ν + 1 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["\[Nu]", "+", "1"]], ")"]], "j"], Pochhammer] j ! k = 0 ( - 1 ) j ( j - ν ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "-", "\[Nu]"]], ")"]], "k"], Pochhammer] ( j + ν + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["j", "+", "\[Nu]", "+", "1"]], ")"]], "k"], Pochhammer] ( j + k ) ! k ! 2 j + k z j + 2 m + 1 /; "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 Condition LegendreQ ν z ν PolyGamma ν 1 2 3 2 -1 j 0 -1 j 2 j j -1 Γ j -1 ν 2 -1 Gamma j ν 1 2 -1 z j j 0 Pochhammer -1 ν j Pochhammer ν 1 j j -1 k 0 -1 j Pochhammer j -1 ν k Pochhammer j ν 1 k PolyGamma j k 1 z j j k k 2 j k -1 m 0 Pochhammer 1 2 m Pochhammer 1 m m Pochhammer 3 2 m -1 j 0 Pochhammer -1 ν j Pochhammer ν 1 j j -1 k 0 -1 j Pochhammer j -1 ν k Pochhammer j ν 1 k j k k 2 j k -1 z j 2 m 1 z 1 [/itex]

 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