Tricomi confluent hypergeometric function: Differentiation (formula 07.33.20.0008)

HypergeometricU

$Mathematica Notation$

Hypergeometric Functions

HypergeometricU[a,b,z]

Differentiation

Symbolic differentiation

With respect to b

http://functions.wolfram.com/07.33.20.0008.02

Input Form

Derivative[0, n, 0][HypergeometricU][a, b, z] == Sum[D[Gamma[1 - b]/(Gamma[a - b + 1] Pochhammer[b, k]), {b, n}] ((Pochhammer[a, k] z^k)/k!), {k, 0, Infinity}] + Sum[D[(Pochhammer[a - b + 1, k] Gamma[b - 1] z^(1 - b))/ Pochhammer[2 - b, k], {b, n}] (z^k/(Gamma[a] k!)), {k, 0, Infinity}] /; Element[n, Integers] && n >= 0 && !Element[b, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["0", ",", "n", ",", "0"]], "]"]], "[", "HypergeometricU", "]"]], "[", RowBox[List["a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["D", "[", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", "1"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["b", ",", "k"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["b", ",", "n"]], "}"]]]], "]"]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], SuperscriptBox["z", "k"]]], RowBox[List["k", "!"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["D", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "-", "b", "+", "1"]], ",", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["1", "-", "b"]]]]], RowBox[List[" ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["b", ",", "n"]], "}"]]]], "]"]], FractionBox[SuperscriptBox["z", "k"], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["k", "!"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["b", ",", "Integers"]], "]"]], "]"]]]]]]]]

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> <semantics> <msup> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "n", ",", "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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mfrac> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <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> <mo> ⁢ </mo> <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> </mrow> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> b </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <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> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <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> <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> <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> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> b </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> b </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <semantics> <msup> <mi> U </mi> <mrow> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> n </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox["U", RowBox[List["(", RowBox[List["0", ",", "n", ",", "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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mfrac> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <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> <mo> ⁢ </mo> <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> </mrow> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> b </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <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> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <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> <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> <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> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> b </mi> <mi> n </mi> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> b </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["HypergeometricU", TagBox[RowBox[List["(", RowBox[List["0", ",", "n", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "n"]], "}"]]]]], FractionBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "b"]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", "b", "+", "1"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["b", ",", "k"]], "]"]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"]]], ")"]]]], RowBox[List["k", "!"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["b", ",", "n"]], "}"]]]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "-", "b", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["b", "-", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["1", "-", "b"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]]]]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List["!", RowBox[List["b", "\[Element]", "Integers"]]]]]]]]]]]]

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

2001-10-29