Continued fraction constants are constants defined by infinite continued fractions. Closed forms are available for some generalized continued fractions having particularly simple partial numerators and denominators.
The Ramanujan continued fractions provide a class of continued fraction constants. The Trott constants match their decimal digits to their partial quotients or to interleaved partial numerators and partial denominators. Some constructions allow zero partial quotients.
The first in a series of other famous continued fraction constants is the infinite regular continued fraction
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(1)
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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
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(2)
|
where 
has the initial conditions 
, 
and 
has the initial conditions 
, 
. These can be solved exactly to yield
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 |  | ![2[(-1)^nI_n(2)K_1(2)+I_1(2)K_n(2)]](/images/equations/ContinuedFractionConstants/Inline21.svg) |
(3)
|
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 |  | ![2[(-1)^(n-1)I_n(2)K_0(2)+I_0(2)K_n(2)],](/images/equations/ContinuedFractionConstants/Inline27.svg) |
(4)
|
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
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(5)
|
(OEIS A052119; Lehmer 1973, Rabinowitz 1990; Borwein et al. 2004, p. 35).
The related constant defined by the generalized continued fraction
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(6)
|
has 
th
convergent, for 
, given by
![(A_n)/(B_n)=[(Gamma(n+2))/(!(n+1))-1]^(-1),](/images/equations/ContinuedFractionConstants/NumberedEquation2.svg) |
(7)
|
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
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(8)
|
(OEIS A073333).
Another similar continued fraction constant that can be computed in closed form is
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 |  | ![sqrt(2/(epi))[erfc(2^(-1/2))]^(-1)](/images/equations/ContinuedFractionConstants/Inline66.svg) | |
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(9)
|
(OEIS A111129), where erfc is the complementary error function. For 
, 2, ..., the first few convergents
to 
are 1, 1/3, 2/3, 4/9, 7/12, 19/39, 68/123, ... (OEIS A225435
and A225436). Adding 1 to each gives the corresponding
convergents to 
. The unreduced numerators 
and denominators 
for 
satisfy
 |
(10)
|
for 
,
with 
,
,
and 
.
Another closed-form continued fraction is given by
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(11)
|
(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 ![[b_0;b_1,b_2...]](/images/equations/ContinuedFractionConstants/Inline92.svg)
with partial quotients that are in arithmetic
progression is given by
![[A+D,A+2D,A+3D,...]=(I_(A/D)(2/D))/(I_(1+A/D)(2/D))](/images/equations/ContinuedFractionConstants/NumberedEquation4.svg) |
(12)
|
(Schroeppel 1972) for real 
and 
with 
and 
. These conditions make all the partial quotients positive.
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 (12).
