|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.33.20.0018.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Derivative[1, 0, 0][HypergeometricU][m, n, z] ==
(-(Log[z] + PolyGamma[m])) HypergeometricU[m, n, z] +
(1/(m - 1)!) Sum[(Binomial[n - m - 1, k] (m + k - 1)! PolyGamma[m + k])/
z^(m + k), {k, 0, n - m - 1}] - (1/(m - 1)!)
Sum[Binomial[n - m - 1, k] (k + m - 1)! (-z)^(-k - m)
(Sum[(LaguerreL[k + m - 1 - p, -k - m + p, -z] LaguerreL[p - 1, -p, z])/
p, {p, 1, k + m - 1}] - E^z (-CoshIntegral[z] + SinhIntegral[z])
LaguerreL[k + m - 1, -k - m, -z]), {k, 0, n - m - 1}] /;
Element[m, Integers] && m > 0 && Element[n, Integers] && n > m
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0", ",", "0"]], "]"]], "[", "HypergeometricU", "]"]], "[", RowBox[List["m", ",", "n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "m", "]"]]]], ")"]]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List["m", ",", "n", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["m", "+", "k"]], "]"]]]], SuperscriptBox["z", RowBox[List["m", "+", "k"]]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "-", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["k", "+", "m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "1", "-", "p"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m", "+", "p"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["p", "-", "1"]], ",", RowBox[List["-", "p"]], ",", "z"]], "]"]]]], "p"]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["CoshIntegral", "[", "z", "]"]]]], "+", RowBox[List["SinhIntegral", "[", "z", "]"]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "1"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "m"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <semantics> <mrow> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> ψ </mi> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> U </mi> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[" ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["log", "(", "z", ")"]], "+", RowBox[List[TagBox["\[Psi]", PolyGamma], "(", "m", ")"]]]], ")"]], "U"]]]], HypergeometricU] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> p </mi> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Shi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> Chi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mtext> </mtext> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> > </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msup> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation> </semantics> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <semantics> <mrow> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> ψ </mi> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> U </mi> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[" ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["log", "(", "z", ")"]], "+", RowBox[List[TagBox["\[Psi]", PolyGamma], "(", "m", ")"]]]], ")"]], "U"]]]], HypergeometricU] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "m", "-", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mn> 1 </mn> <mi> p </mi> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> p </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mo> - </mo> <mi> p </mi> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mi> z </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Shi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> Chi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <mi> L </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msubsup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mtext> </mtext> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> > </mo> <mi> m </mi> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["HypergeometricU", TagBox[RowBox[List["(", RowBox[List["1", ",", "0", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["m_", ",", "n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "m", "]"]]]], ")"]]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List["m", ",", "n", ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "k", "-", "1"]], ")"]], "!"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["m", "+", "k"]], "]"]]]], SuperscriptBox["z", RowBox[List["m", "+", "k"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "m", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "m", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "m", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "-", "m"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["k", "+", "m", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "1", "-", "p"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m", "+", "p"]], ",", RowBox[List["-", "z"]]]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["p", "-", "1"]], ",", RowBox[List["-", "p"]], ",", "z"]], "]"]]]], "p"]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["CoshIntegral", "[", "z", "]"]]]], "+", RowBox[List["SinhIntegral", "[", "z", "]"]]]], ")"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["k", "+", "m", "-", "1"]], ",", RowBox[List[RowBox[List["-", "k"]], "-", "m"]], ",", RowBox[List["-", "z"]]]], "]"]]]]]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "m"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| |
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|