A Viskovatov algorithm for Hermite-Pade polynomials

We propose and justify an algorithm for producing Hermite-Pade polynomials of type I for an arbitrary tuple of m + 1 formal power series [f(0), ..., f(m)], m >= 1, about the point z = 0 (f(j) is an element of C[[z]]) under the assumption that the series have a certain (`general position') nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for constructing Pade polynomials (for m = 1 our algorithm coincides with the Viskovatov algorithm). The algorithm is based on a recurrence relation and has the following feature: all the Hermite-Pade polynomials corresponding to the multi-indices (k, k, k, ..., k, k), (k+1, k, k, ..., k, k), (k+1, k+1, k, ..., k, k), ..., (k+1, k+1, k+1, ..., k+1, k) are already known at the point when the algorithm produces the Hermite-Pade polynomials corresponding to the multiindex (k +1, k +1, k+1, ..., k +1, k+1). We show how the Hermite-Pade polynomials corresponding to different multi-indices can be found recursively via this algorithm by changing the initial conditions appropriately. At every step n, the algorithm can be parallelized in m + 1 independent evaluations.