Some Algebraic Properties of Hermite–Padé Polynomials

Let be a family of formal series in nonnegative powers of the variable with the condition . It is assumed that this family is in general position. For the given family of series and -dimensional multi-indices , , constructions are given of Hermite–Padé polynomials of the 1st and 2nd types of degrees and , respectively, with the following property. Let and be two polynomial matrices, , generated by Hermite–Padé polynomials of the 1st and 2nd types corresponding to the multi-indices , . Then the following identity holds: where is the identity matrix. The result is motivated by a number of new applications of the Hermite–Padé polynomials recently arisen in connection with studies of the monodromy properties of Fuchsian systems of differential equations.