In this paper we prove a conjecture on a common region of a convergence of Padé iterations for the matrix sector function. For this purpose we show that all Padé approximants to a special case of hypergeometric function have a power series expansion with positive coefficients. Using a sharpened version of Schwarz’s lemma, we also demonstrate a better estimate of the convergence speed. Our resul...

A numerical method for solving the Flierl–Petviashivili (FP) equation and its variants is proposed. The proposed scheme is based on differential transform method (DTM) and Padé approximants. The DTM-Padé technique introduces an alternative framework designed to overcome the difficulty of the singular point at x = 0. The numerical results demonstrates the validity and applicability of the method...

This work is about Frobenius-Padé approximants for series of orthogonal polynomials of dimension d (d ∈ N). Concerning to the series, we give the projection property of partial sums, we show how to compute their coe cients, and how to get the coe cients of the product of a series by a polynomial. Concerning to the approximants we work essentially about their recursive computation. Also, we give...

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that...

Stieltjes' constants γn are the coefficients in Laurent series for zeta function ζ(s) at pole s=1 (Euler's constant γ0 is term of this expansion). The main result paper construction new sequences approximations any one γn. rests on a generalization “remainder Padé approximants” method introduced by first named author 1996. We also focus γ1 which we construct faster using ordinary remainder appr...

In this paper we show how one can obtain simultaneous rational approximants for ζq(1) and ζq(2) with a common denominator by means of Hermite-Padé approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, ζq(1), ζq(2) are linearly independent over Q. In particular this implies that ζq(1) and ζq(2) are irrational. Furthermor...

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...