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BesselI






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselI[nu,z] > Representations through more general functions > Through Meijer G > Generalized cases for products of Bessel I





http://functions.wolfram.com/03.02.26.0129.01









  


  










Input Form





BesselI[-1 - n - \[Nu], z] BesselI[\[Nu], z] == (-(-1)^Floor[n/2]) Sqrt[2 Pi] MeijerG[{{0, 1/2}, {1/4}}, {{(1 + n)/2}, {\[Nu] + (1 + n)/2, -((1 + n)/2), -((1 + n)/2) - \[Nu], 1/4}}, z, 1/2] + (2/Sqrt[Pi]) Sum[(((-1)^Floor[(1 + n)/2] z^(-1 + 2 k - n) Gamma[1/2 + k - n + Floor[n/2]])/(k! Gamma[k - n - \[Nu]] Gamma[1 + k + \[Nu]])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}] /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "n_", "-", "\[Nu]_"]], ",", "z_"]], "]"]], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", FractionBox["1", "4"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["1", "+", "n"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox[RowBox[List["1", "+", "n"]], "2"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], ",", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]], "-", "\[Nu]"]], ",", FractionBox["1", "4"]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "k"]], "-", "n"]]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "+", "\[Nu]"]], "]"]]]]]]]]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02