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Higher order recurrences and row sequences of Hermite-Padé approximation. (English) Zbl 1412.30124

The paper gives first a discussion of recurrence relations \(\sum_{k=0}^m \alpha_{n,k}f_n=0\). A solution defines a function \(f(z)=\sum_{n=0}^\infty f_nz^n\) converging in a disk with radius \(R_0(f)\). Suppose the polynomial \(\lim_{n\to\infty} \sum_{k=0}^m\alpha_{n,k}z^k\) exists with zeros \(\lambda_k\), \(k=1,\ldots,m\). Then under the condition that all the \(|\lambda_k|\) are mutually different, a fundamental system of solutions \(\{f^k:k=1,\ldots,m\}\) exists such that \(R_0(f^k)=1/|\lambda_k|\) and \(1/\lambda_k\) is a singularity of \(f^k\). This follows from theorems by Poincaré and Perron. In this paper the authors extend this result to the case where some of the \(\lambda_k\) have an equal modulus. A link with type-II Hermite-Padé approximation (i.e., simultaneous Padé approximation of a vector of formal power series by a vector of rational functions with restricted numerator degrees and a common denominator) allows to apply their result to link the zeros of the common denominator to so-called system poles and their multiplicities.

MSC:

30E10 Approximation in the complex plane
65Q30 Numerical aspects of recurrence relations
41A21 Padé approximation
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References:

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