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http://functions.wolfram.com/03.21.23.0001.01
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Sum[(SphericalBesselJ[k + \[Nu], x]/k!) x^k, {k, 0, Infinity}] ==
Sqrt[Pi/2] (1/Sqrt[x]) BesselI[1/2 + \[Nu], x]
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["k", "+", "\[Nu]"]], ",", "x"]], "]"]]]], RowBox[List["k", "!"]]], SuperscriptBox["x", "k"]]]]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], FractionBox["1", SqrtBox["x"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "x"]], "]"]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mrow> <msub> <mi> j </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mi> k </mi> </msup> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo>  </mo> <mrow> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <mfrac> <mrow> <mtext> </mtext> <mn> 1 </mn> </mrow> <msqrt> <mi> x </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mrow> <mi> ν </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> k </ci> <ci> ν </ci> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> x </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <ci> ν </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["k_", "+", "\[Nu]_"]], ",", "x_"]], "]"]], " ", SuperscriptBox["x_", "k_"]]], RowBox[List["k_", "!"]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["BesselI", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", "x"]], "]"]]]], SqrtBox["x"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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