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 $[f_0,…,f_m]$, $m\geq1$, about the point $z=0$ ($f_j\in\mathbb{C}[[z]]$) 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 multi-index $(k+1,k+1,k+1,…,k+1,k+1)$.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. Bibliography: 30 titles.

formal power series, Hermite-Pade polynomials, Viskovatov algorithm