Continued Fraction Constant

A continued fraction with partial quotients which increase in Arithmetic Progression is

$$[A+D, A+2D, A+3D, \ldots]={I_{A/D}\left({2\over D}\right)\over I_{1+A/D}\left({2\over D}\right),}$$

where $I_n(x)$ is a Modified Bessel Function of the First Kind (Beeler et al. 1972, Item 99). A special case is

$$C=0+{1\over\strut\displaystyle 1+{1\over\strut\displaystyle ... ...strut\displaystyle 4+{1\over\strut\displaystyle 5+\ldots}}}}},$$

which has the value

$$C={I_1(2)\over I_0(2)}=0.697774658\ldots$$

(Lehmer 1973, Rabinowitz 1990).

See also e, Golden Mean, Modified Bessel Function of the First Kind, Pi, Rabbit Constant, Thue-Morse Constant

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html

Guy, R. K. ``Review: The Mathematics of Plato's Academy.'' Amer. Math. Monthly 97, 440-443, 1990.

Lehmer, D. H. ``Continued Fractions Containing Arithmetic Progressions.'' Scripta Math. 29, 17-24, 1973.

Rabinowitz, S. Problem E3264. ``Asymptotic Estimates from Convergents of a Continued Fraction.'' Amer. Math. Monthly 97, 157-159, 1990.