We give a short introduction to Padé approximation (rational approximation to a function with close contact at one point) and to Hermite-Padé approximation (simultaneous rational approximation to several functions with close contact at one point) and show how orthogonality plays a crucial role. We give some insight into how logarithmic potential theory helps in describing the asymptotic behavio...

ABSTRACT Among various analytic approximations for the growth of density fluctuations in the expanding Universe, Zel’dovich approximation and its extensions in Lagrangian scheme are known to be accurate even in weakly non-linear regime. The aim of this paper is to investigate the reason why these Zel’dovich-type approximations work accurately beyond the linear regime from the following two poin...

A modified Padé approximant is used to construct an equation of state, which has the same large-density asymptotic behavior as the model fluid being described, while still retaining the low-density behavior of the virial equation of state (virial series). Within this framework, all sequences of rational functions that are analytic in the physical domain converge to the correct behavior at the s...

Padé approximants are used to find approximate vortex solutions of any winding number in the context of Gross-Pitaevskii equation for a uniform condensate and condensates with axisymmetric trapping potentials. Rational function and generalised rational function approximations of axisymmetric solitary waves of the Gross-Pitaevskii equation are obtained in two and three dimensions. These approxim...

Ahsrruct: Section 1 describes the univariate situation in the case of non-normal Pade approximants and Cordellier’s extension of the famous five-star identity of Wynn. Section 2 repeats our definition of multivariate Pade approximants and proves a number of theorems that remain valid when going from the univariate to the multivariate case. These theorems and more new results given in Section 3....

We present the pseudo-ǫ expansions (τ -series) for the critical exponents of a λφ4 threedimensional O(n)-symmetric model obtained on the basis of six-loop renormalization-group expansions. Concrete numerical results are presented for physically interesting cases n = 1, n = 2, n = 3 and n = 0, as well as for 4 ≤ n ≤ 32 in order to clarify the general properties of the obtained series. The pseudo...

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Padé approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Padé approximant...

Terms up to order (Γ/kBT)5 in the high-temperature expansion of the orientational specific heat of orthopara alloys of solid H2 or D2 are evaluated. Good agreement is obtained between theory and experiment using a Padé approximant and effective values of the quadrupolar coupling constant, Γeff/Γ0=0.83 for D2 and Γeff/Γ0=0.80 for H2, where Γ0 is the value for a rigid lattice. These values agree ...