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A334632 Decimal expansion of Sum_{k>=0} (-1)^k / ((2*k)!)^2.

%I A334632 #21 Jan 05 2025 09:39:57
%S A334632 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 A334632 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 A334632 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 A334632 Decimal expansion of Sum_{k>=0} (-1)^k / ((2*k)!)^2.
%D A334632 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 55, page 552.
%H A334632 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KelvinFunctions.html">Kelvin Functions</a>.
%H A334632 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kelvin_functions">Kelvin functions</a>.
%F A334632 Equals Re(BesselJ(0, 2*exp(3*Pi*i/4))).
%e A334632 0.75173418271380822855099890123074657595958657660729200273884...
%p A334632 evalf(Sum((-1)^k/(2*k)!^2, k=0..infinity), 120);
%t A334632 RealDigits[KelvinBer[0, 2], 10, 120][[1]]
%t A334632 RealDigits[Re[Hypergeometric0F1Regularized[1, I]], 10, 120][[1]]
%t A334632 RealDigits[HypergeometricPFQ[{}, {1/2, 1/2, 1}, -1/16], 10, 120][[1]] (* _Vaclav Kotesovec_, Jul 19 2021 *)
%o A334632 (PARI) sumalt(k=0, (-1)^k/(2*k)!^2)
%Y A334632 Cf. A334379.
%K A334632 nonn,cons
%O A334632 0,1
%A A334632 _Vaclav Kotesovec_, Sep 10 2020