On Some Algebraic Properties of Hermite--Padé Polynomials

Abstract

Let [f0, . . . , fm] be a tuple of series in nonnegative powers of 1/z, fj(∞) = 0. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite-Padé polynomials to the given tuple of degrees n and mn respectively and the corresponding (m + 1)-multi-indexes with the following property. Let M1(z) and M2(z) be two (m + 1) × (m + 1) polynomial matrices, M1(z), M2(z) ∈ GL(m + 1, C[z]), generated by type I and type II Hermite-Padé polynomials respectively. Then we have M1(z)M2(z) ≡ Im+1, where Im+1 is the identity (m + 1) × (m + 1)matrix.The result is motivated by some novel applications of Hermite-Padé polynomials to the investigation of monodromy properties of Fuchsian systems of differential equations; see [12], [5], [10], [6], [8].Bibliography: [12] titles.