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http://functions.wolfram.com/07.33.20.0003.01
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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]
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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"]], "]"]], "]"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <semantics> <msup> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]]], HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msup> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </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> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <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> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> b </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <semantics> <msup> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "1", ",", "0"]], ")"]]], HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> 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] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msup> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </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> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b", "+", "1"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["2", "-", "b"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <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> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> b </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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