The term "continued fraction" is used to refer to a class of expressions of which generalized continued fraction
of the form
(and the terms may be integers, reals, complexes, or functions of these) are the most general variety (Rocket and Szüsz 1992, p. 1).
Wallis first used the term "continued fraction" in his Arithmetica infinitorum of 1653 (Havil 2003, p. 93), although other sources list the publication date
as 1655 or 1656. An archaic word for a continued fraction is anthyphairetic ratio.
The most successful algorithm employed by the Ramanujan Project (Raayoni et al. 2021) relied on a brute-force search over the space of polynomial continued fractions
to find new formulas for mathematical constants. Elimelech et al. (2023) subsequently
used algorithm involving factorial reduction
to search for new polynomial continued fraction formulas, discovering hundreds of
new formulas for mathematical constants, including , , , and .
Ben David, N.; Nimri, G.; Mendlovic, U.; Manor, Y.; and Kaminer, I. "On the Connection Between Irrationality Measures and Polynomial
Continued Fractions." 5 Nov 2021. https://arxiv.org/abs/2111.04468.Cuyt,
A. A.; Petersen, V.; Verdonk, B.; Waadeland, H.; and Jones, W. B. Handbook
of Continued Fractions for Special Functions. Dordrecht, Netherlands: Springer,
2008. Elimelech, R.; David, O.; De la Cruz Mengual, C.; Kalisch, R.; Berndt, W.;
Shalyt, M.; Silberstein, M.; Hadad, Y.; and Kaminer, I. "Algorithm-Assisted
Discovery of an Intrinsic Order Among Mathematical Constants." 22 Aug 2023.
https://arxiv.org/abs/2308.11829.Havil,
J. Gamma:
Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.Raayoni,
G; Gottlieb, S.; Manor, Y.; Pisha, G.; Harris, Y.; Mendlovic, U.; Haviv, D.; Hadad,
Y.; and Kaminer, I. "Generating Conjectures on Fundamental Constants With the
Ramanujan Machine." Nature590, 67-73, 2021.Rockett,
A. M. and Szüsz, P. Continued
Fractions. New York: World Scientific, 1992.
for all 
, leaving
is an integer and the remainder of the partial
denominators 
for 
are positive integers, the continued fraction is known as a regular
continued fraction.
, 
, 
, and 
.
