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 ParabolicCylinderD

 http://functions.wolfram.com/07.41.20.0004.01

 Input Form

 Derivative[1, 0][ParabolicCylinderD][2 n, z] == (-(1/2)) ParabolicCylinderD[2 n, z] Log[2] + n! Pochhammer[1/2, n] ((2^(-1 + 2 n)/(2 n)!) ParabolicCylinderD[2 n, z] (-EulerGamma + PolyGamma[1/2 - n] + Pi Erfi[z/Sqrt[2]] - z^2 HypergeometricPFQ[{1, 1}, {3/2, 2}, z^2/2]) + (((-1)^n 2^(-(3/2) + n))/Pochhammer[1/2, n]) E^(z^2/4) Sum[(1/(1 + k)) (Sqrt[Pi] z LaguerreL[k, -(1/2) - k, -(z^2/2)] - ((2^(-(1/2) - k) Pi z^(2 + 2 k))/k!) LaguerreL[-(1/2), 1/2 + k, -(z^2/2)]) LaguerreL[-1 - k + n, 1/2 + k, z^2/2], {k, 0, n - 1}] - ((-1)^n 2^(n - 1) Sum[(PolyGamma[1/2 - k] (-2)^k z^(2 k))/ ((n - k)! (2 k)!), {k, 0, n}])/E^(z^2/4)) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "ParabolicCylinderD", "]"]], "[", RowBox[List[RowBox[List["2", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["2", "n"]], ",", "z"]], "]"]], " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List[RowBox[List["n", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]]], RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["2", "n"]], ",", "z"]], "]"]], RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Erfi", "[", FractionBox["z", SqrtBox["2"]], "]"]]]], "-", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2"]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "n"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "4"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox["1", RowBox[List["1", "+", "k"]]], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["LaguerreL", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "2"]]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]]], " "]], RowBox[List["k", "!"]]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "2"]]]]], "]"]]]]]], ")"]], RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "k", "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["n", "-", "1"]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], "k"], SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 D TagBox["D", BesselK] 2 n ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( z ) n ! ( 1 2 ) n TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "n"], Pochhammer] ( 2 2 n - 1 ( 2 n ) ! ( - z 2 2 F 2 ( 1 , 1 ; 3 2 , 2 ; z 2 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] + π erfi ( z 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - n ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) D TagBox["D", ParabolicCylinderD] 2 n ( z ) - ( - 1 ) n 2 n - 1 - z 2 4 k = 0 n ψ TagBox["\[Psi]", PolyGamma] ( 1 2 - k ) ( - 2 ) k z 2 k ( n - k ) ! ( 2 k ) ! + ( - 1 ) n 2 n - 3 2 ( 1 2 ) n TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "n"], Pochhammer] z 2 4 k = 0 n - 1 1 k + 1 ( π z L k - k - 1 2 ( - z 2 2 ) - 2 - k - 1 2 π z 2 k + 2 k ! L - 1 2 k + 1 2 ( - z 2 2 ) ) L - k + n - 1 k + 1 2 ( z 2 2 ) ) - 1 2 D TagBox["D", ParabolicCylinderD] 2 n ( z ) log ( 2 ) /; n Condition 1 0 Subscript BesselK D 2 n z n Pochhammer 1 2 n 2 2 n -1 2 n -1 -1 z 2 HypergeometricPFQ 1 1 3 2 2 z 2 2 -1 Erfi z 2 1 2 -1 PolyGamma 1 2 -1 n -1 ParabolicCylinderD 2 n z -1 -1 n 2 n -1 -1 z 2 4 -1 k 0 n PolyGamma 1 2 -1 k -2 k z 2 k n -1 k 2 k -1 -1 n 2 n -1 3 2 Pochhammer 1 2 n -1 z 2 4 -1 k 0 n -1 1 k 1 -1 1 2 z LaguerreL k -1 k -1 1 2 -1 z 2 2 -1 -1 2 -1 k -1 1 2 z 2 k 2 k -1 LaguerreL -1 1 2 k 1 2 -1 z 2 2 -1 LaguerreL -1 k n -1 k 1 2 z 2 2 -1 -1 1 2 ParabolicCylinderD 2 n z 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["ParabolicCylinderD", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["2", " ", "n_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", "z"]], "]"]], " ", RowBox[List["Log", "[", "2", "]"]]]], "+", RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "n"]], "]"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Erfi", "[", FractionBox["z", SqrtBox["2"]], "]"]]]], "-", RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", "2"]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", "n"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "+", "n"]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "4"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["\[Pi]"], " ", "z", " ", RowBox[List["LaguerreL", "[", RowBox[List["k", ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "2"]]]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "k"]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", "+", RowBox[List["2", " ", "k"]]]]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "2"]]]]], "]"]]]], RowBox[List["k", "!"]]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "k", "+", "n"]], ",", RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], RowBox[List["1", "+", "k"]]]]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "-", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Contributed by

 Brychkov Yu.A. (2006)

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

 2007-05-02