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A070910 Decimal expansion of BesselI(0,2). 38

%I #46 Dec 16 2022 09:03:07

%S 2,2,7,9,5,8,5,3,0,2,3,3,6,0,6,7,2,6,7,4,3,7,2,0,4,4,4,0,8,1,1,5,3,3,

%T 3,5,3,2,8,5,8,4,1,1,0,2,7,8,5,4,5,9,0,5,4,0,7,0,8,3,9,7,5,1,6,6,4,3,

%U 0,5,3,4,3,2,3,2,6,7,6,3,4,2,7,2,9,5,1,7,0,8,8,5,5,6,4,8,5,8,9,8,9,8,4,5,9

%N Decimal expansion of BesselI(0,2).

%H Michael Penn, <a href="https://www.youtube.com/watch?v=-UhFu0g9740">An exponential trigonometric integral.</a>, YouTube video, 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialSums.html">Factorial Sums</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html">Modified Bessel Function of the First Kind</a>.

%F Equals Sum_{k>=0} 1/k!^2.

%F From _Peter Bala_, Aug 19 2013: (Start)

%F Continued fraction expansion: 1/(1 - 1/(2 - 1/(5 - 4/(10 - 9/(17 - ... - (n-1)^2/(n^2+1 - ...)))))). See A006040. Cf. A096789.

%F This continued fraction is the particular case k = 0 of the result BesselI(k,2) = Sum_{n = 0..infinity} 1/(n!*(n+k)!) = 1/(k! - k!/((k+2) - (k+1)/((2*k+5) - 2*(k+2)/((3*k+10) - ... - n*(n+k)/(((n+1)*(n+k+1)+1) - ...))))). See the remarks in A099597 for a sketch of the proof. (End)

%F From _Amiram Eldar_, May 29 2021: (Start)

%F Equals (1/e^2) * Sum_{k>=0} binomial(2*k,k)/k! = e^2 * Sum_{k>=0} (-1)^k*binomial(2*k,k)/k!.

%F Equal (1/(2*Pi)) * Integral_{x=0..2*Pi} exp(2*sin(x)) dx. (End)

%F Equals BesselJ(0,2*i). - _Jianing Song_, Sep 18 2021

%e 2.279585302336...

%t RealDigits[ BesselI[0, 2], 10, 110] [[1]] (* _Robert G. Wilson v_, Jul 09 2004 *)

%t (* Or *) RealDigits[ Sum[ 1/(n!n!), {n, 0, Infinity}], 10, 110][[1]]

%o (PARI) besseli(0,2) \\ _Charles R Greathouse IV_, Feb 19 2014

%Y Cf. A096789, A070913 (continued fraction), A006040.

%Y Bessel function values: A334380 (J(0,1)), A334383 (J(0,sqrt(2)), A091681 (J(0,2)), A197036 (I(0,1)), A334381 (I(0,sqrt(2)), this sequence (I(0,2)).

%K cons,easy,nonn

%O 1,1

%A _Benoit Cloitre_, May 20 2002

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Last modified March 29 02:13 EDT 2024. Contains 371264 sequences. (Running on oeis4.)