A222471 Decimal expansion of the negative of the limit of the continued fraction 1/(1-2/(2-2/(3-2/(4-... in terms of Bessel functions.

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

Offset: 1

Author

Wolfdieter Lang, Mar 23 2013

Keywords

• nonn
• cons

Comments

The continued fraction (0 + K_{k=1..oo} (-2/k))/(-2) = 1/(1-2/(2-2/(3-2/(4- ... converges, and its negative limit is given in the formula section in terms of Bessel functions.

In general, the continued fraction 0 + K_{k=1..oo} (x/k) = x/(1+x/(2+x/(3+... has n-th approximation x*Phat(n,x)/Q(n,x), with the row polynomials Phat of A221913 and Q of A084950. These polynomials are written in terms of Bessel function. Divided by n! = Gamma(n+1) one knows the limit for n -> infinity for these two polynomial systems for given x. This results in the formula 0 + K_{k=1..oo} (x/k) = sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)).

For x=1 see for the limit of the continued fraction A052119 and for the n-th approximation A001053(n+1)/A001040(n+1).

Examples

			-1.4397493218702328058...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A052119 (x=1), A222466 (x=2), A222469/A222470.

Programs

Mathematica

RealDigits[BesselJ[1, 2*Sqrt[2]]/(Sqrt[2]*BesselJ[0, 2*Sqrt[2]]), 10, 50][[1]] (* G. C. Greubel, Aug 16 2017 *)

PARI

besselj(1,sqrt(8))/besselj(0,sqrt(8))/sqrt(2) \\ Charles R Greathouse IV, Feb 19 2014

Formula

Equals (1/2)*sqrt(2)*BesselJ(1,2*sqrt(2))/BesselJ(0,2*sqrt(2)).