In this section, sequence transformations will be discussed which are particularly well suited for the summation of strongly divergent alternating series as they for instance occur in special function theory or in quantum mechanical perturbation theory. Alternative sequence transformations, which can be applied in the case of other convergence acceleration and summation problems, are for instance described in [12,19,23,42].
Normally, the remainders of the partial sums of a strongly divergent
series depend on the index n in a very complicated way. Consequently,
the elimination of 
from 
can be very difficult.
A considerable simplification can often be accomplished by means of a suitable reformulation. Let us consider the following model sequence (see section 3.2 of [23]):
Here, 
is a remainder estimate, which has to be chosen
according to some rule and which may depend on n in a very complicated
way, and 
is a correction term, which should be chosen in
such a way that it depends on n in a relatively smooth way. Moreover,
the products 
should yield sufficiently accurate
approximations to the remainders 
of the sequence to be
transformed.
The principal advantage of the approach based on the model sequence
(4.1) is that now only the correction terms
have to be determined and eliminated, but not
the actual remainders 
. The use of remainder
estimates 
is also an efficient way of
incorporating additional information about the remainders into
the approximation scheme.
The model sequence (4.1) has another undisputable
advantage: There is a systematic way of constructing a sequence
transformation which is exact for this model sequence. Let us assume
that a linear operator 
can be found, which
annihilates the correction term 
, i.e., which satisfies 
. If such a linear annihilation operator 
is known,
a sequence transformation, which is exact for the model sequence
(4.1), can be constructed quite easily. Just apply 
to 
. Since 
annihilates 
and is by assumption linear, a sequence transformation
results which is exact for the model sequence
(4.1) (see eq. (3.2-11) of [23]):
This annihilation operator approach, which was introduced in
[23, section 3.2,] in connection with the sequence transformations
discussed in this section, was recently used by Brezinski and Redivo
Zaglia [43,44] and by Brezinski and Matos
[45] also in the case of other sequence
transformations
.
There are various possible ways of choosing the correction terms 
(see for instance sections 7 - 9 of
[23]). Particularly simple and at the same time very powerful
sequence transformations are obtained if the annihilation operators are
based upon the finite difference operator 
defined by 
(see sections 7 - 9 of [23]). As is
well known, the k-th power of 
annihilates every polynomial
of degree k - 1 in n. Let us now assume that the
correction terms 
are chosen in such a way
that multiplication of 
by some suitable quantity 
yields
a polynomial 
of degree k-1 in n according to
Then, a suitable annihilation operator for 
is the weighted
difference operator 
, and the corresponding
sequence transformation (4.2) is given by the ratio
Different sequence transformations are obtained by specializing 
. For instance, if we choose 
with
, we obtain Levin's sequence transformation [38]:
Here, the same notation as in [23] is used. The shift parameter
has to be positive in order to admit n = 0 in eq. (4.5).
The most obvious choice would be 
. According to Smith and Ford
[46,47], who analyzed the performance of various sequence
transformations in convergence acceleration and summation processes,
Levin's transformation is among the most powerful and also most
versatile sequence transformations that are currently known.
The sequence transformation 
is
by construction exact for the model sequence
Thus, 
is a polynomial of degree k-1 in 
.
Consequently, Levin's transformation should work well if the ratio 
can be expressed as a power series in 
according to
However, a power series in 
is not the only possibility of
representing the ratio 
. An alternative approach
would be to assume that the ratio 
can be expressed
as a factorial series. Let 
be a function which
assumes a constant value as 
. A factorial series for
is an expansion of the following kind:
Here, 
is a Pochhammer symbol.
Factorial series have a long tradition in mathematics. For instance, a
large part of Stirling's book [2], which was first published
in 1730, deals with factorial series. A fairly complete survey of the
older literature on this subject can be found in books by Nielsen
[48] and Nörlund [49]. Since it is extremely easy to
apply higher powers of the difference operator 
to a factorial
series, their properties are discussed in the classic books on finite
differences [49,50,51].
In the context of the summation of divergent series, the books by Borel [52] and Doetsch [53] and the review article by Thomann [54] are of interest, since the connection between factorial series and summability is discussed there. Summation methods, which are closely related to factorial series, are discussed by Gunson and Ng [55,56], and the analytical extension of functions defined by factorial series is discussed by Hughes [57].
In recent years, only few books and articles dealing with factorial series were published. Notable exceptions are a book by Wasow [58], which contains a chapter on factorial series, and articles by Iseki and Iseki [59], by Ramis and Thomann [60], and by Dunster and Lutz [61].
Power series and factorial series have different convergence properties. Power series converge in circles, which may shrink to a single point or extend to contain the whole complex plane, whereas factorial series converge according to Landau [62] in half-planes.
The different convergence properties of power series and factorial
series are demonstrated by the following two infinite series which both
have the same numerical coefficients 
:
The power series diverges for all 
, whereas the
factorial series converges for all x > 0.
Because of the different convergence properties of factorial and power
series it may happen that a given function 
, which possesses
a representation in terms of a divergent asymptotic power series
in 
, possesses also a representation as a convergent
factorial series. The algebraic processes, by means of which the inverse
power series and the factorial series can be transformed into each
other, were already described by Stirling [2] in 1730. A more
modern description of Stirling's method can be found in Nielsen's book
[48, pp. 272 - 282,]. A detailed investigation of the problems
associated with the transformation of an asymptotic series into a
convergent factorial series can be found in a long article by Watson
[63].
The different convergence properties of power series and factorial series had actually motivated me to look for a factorial series analogue of Levin's transformation (4.5).
To derive a sequence transformation, which is based on factorial series,
we only have to assume that the weights 
in eq. (4.4)
are Pochhammer symbols according to 
with 
. This yields the
following sequence transformation (see eq. (8.2-7) of [23]),
which is by construction exact for the model sequence (see eq. (8.2-1) of [23])
Hence, 
is a truncated factorial series. This indicates that 
should give good results if the
ratio 
can be expressed as a factorial series
according to
As in Levin's transformation (4.5), the shift parameter 
has to be positive, and 
is again the most obvious choice.
The numerator and denominator sums of the sequence transformations (4.5) and (4.11) can be computed via homogeneous three-term recursions (compare sections 7.2 and 8.3 of [23]).
Other sequence transformations, which are also special cases of the general sequence transformation (4.4), can be found in sections 7 - 9 of [23] or in section 2.7 of [19]. A sequence transformation, which interpolates between the sequence transformations (4.5) and (4.11), was described in [31].
