►The coefficients of the power series of , and also , are the same until the terms in and , respectively.
►Here for , for , and for and .
►where is the unique root of the equation in the interval , and .
Cody et al. (1971) gives rational approximations for
in the form of quotients of polynomials or quotients of
Chebyshev series. The ranges covered are ,
, , . Precision is
varied, with a maximum of 20S.
Antia (1993) gives minimax rational approximations for
, where is the Fermi–Dirac integral
(25.12.14), for the intervals and
. For each there
are three sets of approximations, with relative maximum errors
is the number of permutations of with cycles of length 1, cycles of length 2, , and cycles of length :
… is the number of set partitions of with subsets of size 1, subsets of size 2, , and subsets of size :
…For each all possible values of are covered.
►where the summation is over all nonnegative integers such that .
►If denotes the right-hand side of (24.19.1) but with the second product taken only for , then for .
►For other information see Chellali (1988) and Zhang and Jin (1996, pp. 1–11).
►For number-theoretic applications it is important to compute for ; in particular to find the irregular pairs
for which .
A method related to “Stickelberger codes” is applied in
Buhler et al. (2001); in particular, it allows for an efficient search for
the irregular pairs . Discrete Fourier transforms are used in the
computations. See also Crandall (1996, pp. 120–124).