Higher-order QCD corrections to H → b$$ \overline{b} $$ from rational approximants

We use rational approximants to study missing higher orders in the massless scalar-current quark correlator. We predict the yet unknown six-loop coefficient of its imaginary part, related to Γ(H → b$$ \overline{b} $$ b ¯ ), to be c5 = −6900 ± 1400. With this result, the perturbative series becomes almost insensitive to renormalization scale variations and the intrinsic QCD truncation uncertainty is tiny. This confirms the expectation that higher-order loop computations for this quantity will not be required in the foreseeable future, as the uncertainty in Γ(H → b$$ \overline{b} $$ b ¯ ) will remain largely dominated by the Standard Model parameters.

Complete Radiation Boundary Conditions for Maxwell’s Equations

We describe the construction, analysis, and implementation of arbitrary-order local radiation boundary condition sequences for Maxwell’s equations. In particular we use the complete radiation boundary conditions which implicitly apply uniformly accurate exponentially convergent rational approximants to the exact radiation boundary conditions. Numerical experiments for waveguide and free space problems using high- order discontinuous Galerkin spatial discretizations are presented.

Two-dimensional generalized instantaneous images and approximations of the Pade type function of two variables

Generalized instantaneous image were introduced by V.K. Dzyaduk [1] in 1981 and proved to be a convenient tool for constructing and studying the Padé approximants and their generalizations (see [2]). The method of generalized instantaneous images proposed by Dzyadyk made it possible to construct and study rational Padé approximants and their generalizations for many classes of special functions from a single position. As an example, the Padé approximants is constructed for a class of basic hypergeometric series, which includes a q-analogue of the exponential function. In this paper the construction of the Pade approximants for the function of two variables is investigated. A two-dimensional functional sequence is constructed, which has a generalized instantaneous image, and rational approximants are determined, which will be generalizations of one-dimensional Padé approximants. The function of the two variables is entirely related to the basic hypergeometric series.

On Infinitely Many Rational Approximants to ζ(3)

A set of second order holonomic difference equations was deduced from a set of simultaneous rational approximation problems. Some orthogonal forms involved in the approximation were used to compute the Casorati determinants for its linearly independent solutions. These solutions constitute the numerator and denominator sequences of rational approximants to ζ ( 3 ) . A correspondence from the set of parameters involved in the holonomic difference equation to the set of holonomic bi-sequences formed by these numerators and denominators appears. Infinitely many rational approximants can be generated.

Some properties of approximants for branched continued fractions of the special form with positive and alternating-sign partial numerators

The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables. Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence. We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants to the same limit.

Phenomenological applications of rational approximants

We illustrate the powerfulness of Padé approximants (PAs) as a summation method and explore one of their extensions, the so-called quadratic approximant (QAs), to access both space- and (low-energy) time-like (TL) regions. As an introductory and pedagogical exercise, the function [Formula: see text] is approximated by both kind of approximants. Then, PAs are applied to predict pseudoscalar meson Dalitz decays and to extract [Formula: see text] from the semileptonic [Formula: see text] decays. Finally, the [Formula: see text] vector form factor in the TL region is explored using QAs.