As already remarked, the asymptotic condition (5.2) can be the starting point for the construction of remainder estimates . In my opinion, it should be possible to derive many interesting results which could lead via the sequence transformations (4.5) and (4.11) to powerful approximation schemes for functions defined by slowly convergent or divergent series.

For instance, in [29] it was shown that the asymptotic series for the modified Bessel function of the second kind,

which diverges quite strongly for all arguments , can be summed efficiently by the sequence transformations and . These two transformations are based upon the remainder estimate (5.3), which corresponds to the first term of the series not contained in the partial sum. It would be interesting to find out whether and how well the efficiency of the ``parent'' sequence transformations (4.5) and (4.11) could be increased by better truncation error estimates.

Special results of that kind would certainly be interesting. However, it
would be much more interesting to construct remainder estimates which
are * generally valid* for whole classes of functions. Of course,
this would probably be much more difficult. Nevertheless, I hope that
the derivation of general results could at least in the case of
Stieltjes series be accomplished.

A function is called * Stieltjes function* if it can be
expressed as a Stieltjes integral according to

Here, is a positive measure on , which assumes infinitely many different values on and which has for all finite and positive moments defined by

The formal series expansion for , which need not be convergent,

is called a * Stieltjes* series if its coefficients are
moments of a positive measure on according
to eq. (6.3):

The classic example of a Stieltjes function with a strongly divergent Stieltjes series is the Euler integral (2.6) and its associated asymptotic series, the Euler series (2.7).

The series (5.16) for , which was summed in Table 3, is also a Stieltjes series. This follows from the integral representation

We only have to set for and for . The moments of this positive measure are given by

Stieltjes functions and their associated Stieltjes series are very important in the theory of divergent series, since they possess a highly developed representation and convergence theory [25,71,26,72,73,74].

Moreover, Stieltjes functions and Stieltjes series are also of considerable importance in quantum mechanical perturbation theory. An example is the quartic anharmonic oscillator which is described by the Hamiltonian

Simon [75] could show that the perturbation series for an energy eigenvalue of the quartic anharmonic oscillator oscillator,

which diverges quite strongly for every nonzero coupling constant , is the negative of a Stieltjes series.

If the positive measure corresponding to the Stieltjes function could be determined directly from the Stieltjes moments , the value of could at least in principle be computed via the integral representation (6.2) even if the Stieltjes series diverges. For instance, the measure corresponding to the asymptotic series (6.1) for the modified Bessel function can be found quite easily [29, eq. (2.25),].

Moreover, there is an extensive literature on moment problems and their
solution [76]. * Necessary* and * sufficient*
conditions, which for instance guarantee that Padé approximants are
able to sum a divergent Stieltjes series, have been known for long. An
example is the condition that all Hankel determinants formed from the
moments have to be strictly positive [26, Theorem
5.1.2,]. Unfortunately, the practical application of this and
similar other conditions is by no means simple, in particular if only
the numerical values of a finite number of Stieltjes moments ,
, ..., are available.

However, there is a comparatively simple * sufficient* condition,
the so-called * Carleman* condition. If the moments
satisfy

then the corresponding moment problem possesses a unique solution, and
the sequences of Padé approximants converge for **j > -
1** to the value of the corresponding Stieltjes function as [77, Theorem 12.11f and Corollary 12.11h,]. It can be
shown that the Carleman condition is satisfied if the moments do
not grow faster than as , with **C** being
a suitable positive constant [78, Theorem 1.3,].

The Carleman condition (6.10) implies that the Euler series
(2.7) can be summed by the Padé approximants
with fixed **j > - 1**. However, the numerical results in Tables 2 and 3
demonstrate convincingly that Padé summation need not be the most
efficient way of extracting useful numerical information from the terms
of a divergent Stieltjes series. Consequently, it should be worth while
to investigate also other techniques for the summation of divergent
Stieltjes series.

Wed Sep 13 12:04:01 PDT 1995