stieltjes series

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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 toconstruct remainder estimates which are generally validfor 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,

es Series http://www.apmaths.uwo.ca/~rcorless/AM563/NOTES/report/no

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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


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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

sufficientconditions, which for instance guarantee that Pad 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 numericalvalues of a finite number of Stieltjes moments , , ..., are available.

However, there is a comparatively simple sufficientcondition, the so-called Carleman condition. If the

moments satisfy

then the corresponding moment problem possesses a unique solution, and the sequences of Pad

approximants converge forj > - 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 Pad approximants

with fixedj > - 1. However, the numerical results in Tables 2 and 3 demonstrate convincingly that

Pad 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.

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Next:Remainder Estimates for Up:OPEN PROBLEMS IN ASYMPTOTICS Previous:On the Choice

Rob Corless

Wed Sep 13 12:04:01 PDT 1995


4/24/2012

stieltjes series

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