Continued Fraction Constant
A continued fraction with partial quotients which increase in Arithmetic Progression is
where
is a Modified Bessel Function of the First Kind (Beeler et al. 1972, Item 99). A special case is
which has the value
(Lehmer 1973, Rabinowitz 1990).See also e, Golden Mean, Modified Bessel Function of the First Kind, Pi, RabbitConstant, 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.
- Cayley-Bacharach Theorem
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