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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 &zeta, ( 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.

33 pages, no figures.-- MSC2000 codes: Primary 42C05, 11B37, 11J72, 11M06; Secondary 30B70, 11A55, 11J70, 33C47. Submitted to: Journal of Approximation Theory A new proof of the irrationality of $\zeta(3)$ is given. The orthogonality relation among certain known forms [17] constitutes a novel ingredient used in the present approach. Here, the same sequences of integers obtained in [10] appear. Apéry’s recurrence relation for the sequence of rational approximants to $\zeta(3)$ are constructively obtained. A simultaneous rational approximation problem is used for such purposes. This research was partially supported by ‘Ramón y Cajal’ research program of Spain, the research project MTM2006-13000-C03-02 of the Ministerio de Educación y Ciencia of Spain, and Projects CCG07-UC3M/ESP-3339 and CCG08-UC3M/ESP-4516 from Comunidad Autónoma de Madrid. No publicado

We generalize our earlier results on rational interpolation which were given in Van Barel and Bultheel (this journal, 1990) for the scalar case and in Bultheel and Van Barel (1990) for the vector case when all the interpolation points coincide, to the case of vector data given at arbitrary points that may coincide or not. This is the vector-valued Newton-Pade problem. We give a recursive algorithm which has the important advantage over other algorithms that overcome a singularity in the interpolation table, not even in the nonnormal vector case. It also generates all the information needed to give all solutions of the problem. ispartof: Journal of computational and applied mathematics vol:33 issue:3 pages:331-346 status: published

In this paper the concept of partial Pade approximation, introduced by Claude Brezinski, is generalised to the case of simultaneous rational approximation with common denominator. The use of information about known poles and/or zeros, can lead to approximants with a better numerical behaviour than in the case of ordinary simultaneous Pade approximation.

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. Furthermore we give an upper bound for the measure of irrationality.