Continued Fraction Constants

A number of closed-form constants can be obtained for generalized continued fractions having particularly simple partial numerators and denominators.

The Ramanujan continued fractions provide a fascinating class of continued fraction constants. The Trott constants are unexpected constants whose partial numerators and denominators correspond to their decimal digits (though to achieve this, it is necessary to allow some partial numerators to equal 0).

The first in a series of other famous continued fraction constants is the infinite regular continued fraction

			(1)
			(2)

The first few convergents of the constant are 0, 1, 2/3, 7/10, 30/43, 157/225, 972/1393, 6961/9976, ... (OEIS A001053 and A001040).

Both numerator and denominator satisfy the recurrence relation

(3)

where has the initial conditions , and has the initial conditions , . These can be solved exactly to yield

			(4)
			(5)
			(6)
			(7)

where is a modified Bessel function of the first kind and is a modified Bessel function of the second kind. Therefore, as , the infinite continued fraction is given by

			(8)
			(9)
			(10)

(OEIS A052119; Lehmer 1973, Rabinowitz 1990; Borwein et al. 2004, p. 35).

The related constant defined by the generalized continued fraction

			(11)
			(12)

has th convergent is given by

(13)

where is the gamma function and is the subfactorial. The first few convergents are therefore 1, 1/2, 3/5, 11/19, 53/91, 103/177, ... (OEIS A053557 and A103816). As , this gives the value

			(14)
			(15)

(OEIS A073333).

Another similar continued fraction constant that can be computed in closed form is

			(16)
			(17)
			(18)
			(19)

(OEIS A111129), where erfc is the complementary error function. No closed form is known for the convergents, but for , 2, ..., the first few convergents are 1, 1/3, 2/3, 4/9, 7/12, 19/39, 68/123, ... (OEIS A225435 and A225436).

Another closed-form continued fraction is given by

			(20)
			(21)
			(22)
			(23)

(OEIS A113011). The first few convergents are 5/3, 29/19, 233/151, 2329/1511, 27949/18131, 78257/50767, ... (OEIS A113012 and A113013).

The general infinite continued fraction with partial quotients that are in arithmetic progression is given by

(24)

(Schroeppel 1972) for real and .

Perron (1954-57) discusses continued fractions having terms even more general than the arithmetic progression and relates them to various special functions. He does not, however, appear to specifically consider equation (24).