Stirling's series is the asymptotic series for the log gamma function. The corresponding expansion of the gamma function is
 |
(1)
|
The expansions hold as 
in each closed sector 
with fixed 
. The principal
branch of 
defines the powers of 
,
while 
denotes the analytic log gamma function branch
that is real on the positive real axis (Abramowitz and
Stegun 1972, p. 257). The series have asymptotic
meaning through finite truncations.
The coefficients have 
. For 
, the coefficient 
of 
can be given explicitly by
 |
(2)
|
where 
is the number of permutations of 
objects with 
permutation cycles all
of which have length at least 3 (Comtet 1974, p. 267). Another formula for the
s is given by the recurrence
relation
 |
(3)
|
for 
,
with 
.
Then
 |
(4)
|
where 
is the double factorial (Borwein and Corless
1999).
The series for 
is obtained by adding an additional factor of 
,
 |  |  |
(5)
|
 |  |  |
(6)
|
The corresponding expansion for the log gamma function is
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A046968 and A046969), where 
is a Bernoulli number. Interestingly, while the
numerators in this expansion are the same as those of 
for the first several hundred terms, they differ
at 
,
1185, 1240, 1269, 1376, 1906, 1910, ... (OEIS A090495),
with the corresponding ratios being 37, 103, 37, 59, 131, 37, 67, ... (OEIS A090496).
." Amer. Math. Monthly 97, 826-829, 1990.Morse,
P. M. and Feshbach, H. 
." Proc. Nat. Acad. Sci. U.S.A. 28,
59-62, 1942.Wrench, J. W. Jr. "Concerning Two Series for the
Gamma Function." Math. Comput. 22, 617-626, 1968.