We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that o...

Recent ly McCabe and Murphy have considered the two-po in t Pad6 approx iman t s to a funct ion for which (formal) power series expansions at the origin and at inf ini ty are given. In this paper these approximat ions are slightly modi f ied and de te rminan t representa t ions for them are given. The existence o f various th ree t e rm recursion relations for the numera tors and denominators o...

In this paper, an aproximate analytical method called the differential transform method (DTM) is used as a tool to give approximate solutions of nonlinear oscillators with fractional nonlinearites. The differential transformation method is described in a nuthsell. DTM can simply be applied to linear or nonlinear problems and reduces the required computational effort. The proposed scheme is base...

General T-fractions and M-fractions whose approximants form diagonals in two-point Pad6 tables are subsumed here under the study of Perron-Carath6odory continued fractions (PC-fractions) whose approximants form diagonals in weak two-point Pad6 tables. The correspondence of PCfractions with pairs of formal power series is characterized in terms of Toeplitz determinants. For the subclass of posit...

We compare Monte Carlo results for the interface tension and interface energy of the 3-dimensional Ising model with Padé and inhomogeneous differential approximants of the low temperature series that was recently extended by Arisue to 17 order in u = exp(−4β). The series is expected to suffer from the roughening singularity at u ≈ 0.196. The comparison with the Monte Carlo data shows that the P...

Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function E (x) = ∞ ∑ j1,j2,...,jm=0 x1 1 x j2 2 · · ·x jm m (j1 + j2 + · · ·+ jm)! , the logarithm function L(x) = ∑ j1+j2+···+jm≥1 x1 1 x j2 2 · · ·x jm m j1 + j2 + · · ·+ jm , the Lauricella funct...

Truncated perturbative series (TPS’s) of any observable have the unphysical dependence on the choice of the renormalization scale (RScl). The diagonal Padé approximants (dPA’s) to any TPS of an observable possess the favorable property of being invariant in the large-β0 limit. This means that they are invariant under the change of the RScl μ when the “running” coupling parameter αs(μ ) evolves ...

The classical Lanczos process can be used to efficiently generate Padé approximants of the transfer function of a given single-input single-output time-invariant linear dynamical system. Unfortunately, in general, the resulting reduced-order models based on Padé approximation do not preserve the stability, and possibly passivity, of the original linear dynamical system. In this paper, we descri...

We present a sequent calculus for formally verifying modal μ-calculus properties of concurrent processes. Building on work by Dam and Gurov, the proof system contains rules for the explicit manipulation of fixed-point approximants. We develop a new syntax for approximants, incorporating, in particular, modalities for approximant modification. We make essential use of this feature to prove our m...