A334632 - OEIS

%I #21 Jan 05 2025 09:39:57

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

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

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

%N Decimal expansion of Sum_{k>=0} (-1)^k / ((2*k)!)^2.

%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 55, page 552.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KelvinFunctions.html">Kelvin Functions</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kelvin_functions">Kelvin functions</a>.

%F Equals Re(BesselJ(0, 2*exp(3*Pi*i/4))).

%e 0.75173418271380822855099890123074657595958657660729200273884...

%p evalf(Sum((-1)^k/(2*k)!^2, k=0..infinity), 120);

%t RealDigits[KelvinBer[0, 2], 10, 120][[1]]

%t RealDigits[Re[Hypergeometric0F1Regularized[1, I]], 10, 120][[1]]

%t RealDigits[HypergeometricPFQ[{}, {1/2, 1/2, 1}, -1/16], 10, 120][[1]] (* _Vaclav Kotesovec_, Jul 19 2021 *)

%o (PARI) sumalt(k=0, (-1)^k/(2*k)!^2)

%Y Cf. A334379.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Sep 10 2020