Some Algebraic Properties of Hermite-Pade Polynomials

Let [f(0), ... , f(m)] be a family of formal series in nonnegative powers of the variable 1/z with the condition f(j)(8) ? 0. It is assumed that this family is in general position. For the given family of series and (m + 1)-dimensional multi-indices n(k) ? Nm+1, k = 0, ... , m, constructions are given of Hermite-Pad e' polynomials of the 1st and 2nd types of degrees < n and < mn, respectively, with the following property. Let M1(z) and M2(z) be two (m + 1) x (m + 1) polynomial matrices, M-1(z), M-2(z) ?GL(m + 1, C[z]), generated by Hermite-Pade' polynomials of the 1st and 2nd types corresponding to the multi-indices nk E Nm+1, k = 0, ... , m. Then the following identity holds: M-1(z)M2T (z) = I, M-1(0) = M-2(0) = I, where I is the identity (m + 1) x (m + 1) matrix. The result is motivated by a number of new applications of the Hermite-Pade' polynomials recently arisen in connection with studies of the monodromy properties of Fuchsian systems of differential equations.