Mathematics – Classical Analysis and ODEs
Scientific paper
2004-09-20
Advances in Mathematics vol. 204:2 (2006), 448-480.
Mathematics
Classical Analysis and ODEs
25 pages, 8 figures, LaTeX2e, revised version to appear in Advances in Mathematics
Scientific paper
10.1016/j.aim.2005.06.002
We construct a new scheme of approximation of any multivalued algebraic function $f(z)$ by a sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by $f(z)$. Compared to the usual Pad\'e approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Pad\'e Conjecture and Nuttall's Conjecture for the sequence $\{r_{n}(z)\}_{n\in \mathbb{N}}$ in the complement $\mathbb{CP}^1\setminus \D_{f}$, where $\D_{f}$ is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family $\{r_{n}(z)\}_{n\in \mathbb{N}}$. As an application we settle the so-called 3-conjecture of Egecioglu {\em et al} dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.
Bögvad Rikard
Borcea Julius
Shapiro Boris
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