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 HypergeometricU

 http://functions.wolfram.com/07.33.20.0003.01

 Input Form

 Derivative[0, 1, 0][HypergeometricU][a, b, z] == (PolyGamma[1 + a - b] - Pi Cot[b Pi]) HypergeometricU[a, b, z] - ((z^(1 - b) Gamma[b - 1] Log[z])/Gamma[a]) Hypergeometric1F1[1 + a - b, 2 - b, z] - (Gamma[1 - b]/Gamma[1 + a - b]) Sum[(PolyGamma[b + k] Pochhammer[a, k] z^k)/(Pochhammer[b, k] k!), {k, 0, Infinity}] - ((z^(1 - b) Gamma[b - 1])/Gamma[a]) Sum[(Pochhammer[1 + a - b, k]/(Pochhammer[2 - b, k] k!)) (PolyGamma[1 + a - b + k] - PolyGamma[2 - b + k]) z^k, {k, 0, Infinity}] /; !Element[b, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["0", ",", "1", ",", "0"]], "]"]], "[", "HypergeometricU", "]"]], "[", RowBox[List["a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "-", "b"]], "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]]]]]], ")"]], RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", "b", ",", "z"]], "]"]]]], " ", "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], RowBox[List["Gamma", "[", "a", "]"]]], RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "b"]], ",", RowBox[List["2", "-", "b"]], ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b"]], "]"]], " "]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["b", "+", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["b", ",", "k"]], "]"]], RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]], " "]], RowBox[List["Gamma", "[", "a", "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "b"]], ",", "k"]], "]"]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "-", "b", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "-", "b", "+", "k"]], "]"]]]], ")"]], SuperscriptBox["z", "k"]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["b", ",", "Integers"]], "]"]], "]"]]]]]]

 MathML Form

 U ( 0 , 1 , 0 ) TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]]], HypergeometricU] ( a , b , z ) ( ψ TagBox["\[Psi]", PolyGamma] ( a - b + 1 ) - π cot ( b π ) ) U TagBox["U", HypergeometricU] ( a , b , z ) - Γ ( b - 1 ) z 1 - b log ( z ) Γ ( a ) 1 F 1 ( a - b + 1 ; 2 - b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["a", "-", "b", "+", "1"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", "b"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - Γ ( b - 1 ) z 1 - b Γ ( a ) k = 0 ( a - b + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b", "+", "1"]], ")"]], "k"], Pochhammer] ( ψ TagBox["\[Psi]", PolyGamma] ( a - b + k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( 2 - b + k ) ) z k ( 2 - b ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], "k"], Pochhammer] k ! - Γ ( 1 - b ) Γ ( a - b + 1 ) k = 0 ψ TagBox["\[Psi]", PolyGamma] ( b + k ) ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] z k ( b ) k TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "k"], Pochhammer] k ! /; b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] U ( 0 , 1 , 0 ) TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]]], HypergeometricU] ( a , b , z ) ( ψ TagBox["\[Psi]", PolyGamma] ( a - b + 1 ) - π cot ( b π ) ) U TagBox["U", HypergeometricU] ( a , b , z ) - Γ ( b - 1 ) z 1 - b log ( z ) Γ ( a ) 1 F 1 ( a - b + 1 ; 2 - b ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["a", "-", "b", "+", "1"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", "b"]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - Γ ( b - 1 ) z 1 - b Γ ( a ) k = 0 ( a - b + 1 ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b", "+", "1"]], ")"]], "k"], Pochhammer] ( ψ TagBox["\[Psi]", PolyGamma] ( a - b + k + 1 ) - ψ TagBox["\[Psi]", PolyGamma] ( 2 - b + k ) ) z k ( 2 - b ) k TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], "k"], Pochhammer] k ! - Γ ( 1 - b ) Γ ( a - b + 1 ) k = 0 ψ TagBox["\[Psi]", PolyGamma] ( b + k ) ( a ) k TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] z k ( b ) k TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "k"], Pochhammer] k ! /; b TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["HypergeometricU", TagBox[RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "-", "b"]], "]"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List["a", ",", "b", ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]], ")"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "b"]], ",", RowBox[List["2", "-", "b"]], ",", "z"]], "]"]]]], RowBox[List["Gamma", "[", "a", "]"]]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["b", "+", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["b", ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "b"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "-", "b", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["2", "-", "b", "+", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["Gamma", "[", "a", "]"]]]]], "/;", RowBox[List["!", RowBox[List["b", "\[Element]", "Integers"]]]]]]]]]]

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

 2001-10-29