Sum[BesselI[k, x] t^k, {k, -Infinity, Infinity}] == E^((1/2) x (t + 1/t))

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["BesselI", "[", RowBox[List["k", ",", "x"]], "]"]], " ", SuperscriptBox["t", "k"]]]]], "\[Equal]", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "x", " ", RowBox[List["(", RowBox[List["t", "+", FractionBox["1", "t"]]], ")"]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msub> <mi> I </mi> <mi> k </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> t </mi> <mi> k </mi> </msup> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> x </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> t </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> BesselI </ci> <ci> k </ci> <ci> x </ci> </apply> <apply> <power /> <ci> t </ci> <ci> k </ci> </apply> </apply> </apply> <apply> <exp /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> x </ci> <apply> <plus /> <ci> t </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["BesselI", "[", RowBox[List["k_", ",", "x_"]], "]"]], " ", SuperscriptBox["t_", "k_"]]]]], "]"]], "\[RuleDelayed]", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", "x", " ", RowBox[List["(", RowBox[List["t", "+", FractionBox["1", "t"]]], ")"]]]]]]]]]

2001-10-29