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http://functions.wolfram.com/03.02.23.0005.01
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Sum[((2 k + \[Nu]) BesselI[2 k + \[Nu], x])/(k (k + \[Nu])),
{k, 1, Infinity}] == (-1)^\[Nu] BesselK[\[Nu], x] +
BesselI[\[Nu], x] (Log[x/2] - PolyGamma[\[Nu] + 1]) +
(((-1)^(\[Nu] - 1)/2) \[Nu]! Sum[((-1)^k/((\[Nu] - k) k!)) (x/2)^k
BesselI[k, x], {k, 0, \[Nu] - 1}])/(x/2)^\[Nu] /;
Element[\[Nu], Integers] && \[Nu] >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Nu]"]], ",", "x"]], "]"]]]], RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "\[Nu]"]], ")"]]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox["x", "2"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], "2"], " ", SuperscriptBox[RowBox[List["(", FractionBox["x", "2"], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["\[Nu]", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Nu]", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " "]], RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", FractionBox["x", "2"], ")"]], "k"], " ", RowBox[List["BesselI", "[", RowBox[List["k", ",", "x"]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> ν </mi> <mo> ! </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> x </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> x </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> x </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> ν </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <ci> ν </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <ci> ν </ci> </apply> <ci> x </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <factorial /> <ci> ν </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> BesselI </ci> <ci> k </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <ci> BesselK </ci> <ci> ν </ci> <ci> x </ci> </apply> </apply> <apply> <times /> <apply> <ci> BesselI </ci> <ci> ν </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <times /> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> ν </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k_"]], "+", "\[Nu]_"]], ")"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "k_"]], "+", "\[Nu]_"]], ",", "x_"]], "]"]]]], RowBox[List["k_", " ", RowBox[List["(", RowBox[List["k_", "+", "\[Nu]_"]], ")"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox["x", "2"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["x", "2"], ")"]], RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["\[Nu]", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["\[Nu]", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["x", "2"], ")"]], "k"], " ", RowBox[List["BesselI", "[", RowBox[List["k", ",", "x"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
