ABSTRACT
This chapter discusses approximations to systems of analytic functions in the form of Hermite-Pade rational approximations. These approximations are used to study diophantine approximations of values of functions at particular points. The key method is the computation of consecutive Pade approximants by means of linear recurrences connecting them. Linear recurrences, relating contiguous Pade approximants, are important in arithmetic applications, where they define the asymptotic behavior of sequences of rational approximations to a given number. The chapter describes Mahler's approach on the representation of Pade approximation scheme in terms of matrix recurrences. It presents the corresponding improvements for measures of irrationality of logarithms. The chapter explores the matrix recurrence relations and shows their relations with the algebraic inverse scattering method, associated with completely integrable systems. It also provides the explicit formulas for the recurrences and Pade approximants to other functions of number-theoretical importance are known. The chapter examines the matrix recurrences for Gauss hypergeometric functions.
