Some Algebraic Properties of Hermite–Padé Polynomials

Abstract

Let $[f_0,\dots,f_m]$ be a family of formal series in nonnegative powers of the variable $1/z$ with the condition $f_j(\infty)\ne 0$. It is assumed that this family is in general position. For the given family of series and $(m+1)$-dimensional multi-indices $\mathbf n_k\in\mathbb N^{m+1}$, $k=0,\dots,m$, constructions are given of Hermite–Padé polynomials of the 1st and 2nd types of degrees $\le n$ and $\le mn$, respectively, with the following property. Let $M_1(z)$ and $M_2(z)$ be two $(m+1)\times(m+1)$ polynomial matrices, $M_1(z),M_2(z)\in\operatorname{GL}(m+1,\mathbb C[z])$, generated by Hermite–Padé polynomials of the 1st and 2nd types corresponding to the multi-indices $\mathbf n_k\in\mathbb N^{m+1}$, $k=0,\dots,m$. Then the following identity holds:

$$M_1(z)M_2^{\mathrm T}(z)\equiv I, \qquad M_1(0)=M_2(0)=I,$$

where $I$ is the identity $(m+1)\times(m+1)$ 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.