Distribution of zeros of the Hermite--Pad\'e polynomials for a~system of three functions, and the Nuttall condenser

R. K. Kovacheva ; S. P. Suetin

Informatics and Automation, Function spaces and related problems of analysis, Tome 284 (2014), pp. 176-199

Résumé

The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite–Padé polynomials for a set of $m$ multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) $m$-sheeted Riemann surface possessing certain properties. In this paper, for $m=3$, we introduce a notion of an abstract Nuttall condenser and describe a procedure for constructing (based on this condenser) a three-sheeted Riemann surface $\mathfrak R_3$ that has a canonical decomposition. We consider a system of three functions $\mathfrak f_1,\mathfrak f_2,\mathfrak f_3$ that are rational on the constructed Riemann surface and satisfy the independence condition $\det\bigl[\mathfrak f_k(z^{(j)})\bigr]\not\equiv0$. In the case of $m=3$, we refine the main theorem from Nuttall's paper of 1981. In particular, we show that in this case the complement $\overline{\mathbb C}\setminus B$ of the open (possibly, disconnected) set $B\subset\overline{\mathbb C}$ introduced in Nuttall's paper consists of a finite number of analytic arcs. We also propose a new conjecture concerning strong asymptotic formulas for the Padé approximants.