Abstract

In this article, under certain restrictions, the convergence rate of type II Hermite–Padé approximants (including nondiagonal ones) for a system \(\{{}_{1}F_{1}(1,\gamma;\lambda_{j}z)\}_{j=1}^{k}\), consisting of degenerate hypergeometric functions is found, when \(\{\lambda_{j}\}_{j=1}^{k}\) are different complex numbers, and \(\gamma\in\mathbb{C}\setminus\{0,-1,-2,...\}\). Without the indicated restrictions, similar statements were obtained for approximants of the indicated type, provided that the numbers \(\{\lambda_{j}\}_{j=1}^{k}\) are the roots of the equation \(\lambda^{k}=1\). The theorems proved in this paper complement and generalize the results obtained earlier by other authors.

中文翻译：

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与Mittag-Leffler函数相关的Hermite-Padé近似的渐近性

摘要

在本文中，在某些限制下，系统\（\ {{} _ {1} F_ {1}（1，\ gamma; \ lambda_ {j）的II型Hermite-Padé近似值（包括非对角线近似值）的收敛速度} z）\} _ {j = 1} ^ {k} \），当\（\ {\ lambda_ {j} \} _ {j = 1} ^ {k} \）时，发现由退化的超几何函数组成是不同的复数，和\（\ gamma \ in \ mathbb {C} \ setminus \ {0，-1，-2，... \} \）。如果没有指定的限制，则假设数字\（\ {\ lambda_ {j} \} _ {j = 1} ^ {k} \）是方程\ （\ lambda ^ {k} = 1 \）。本文证明的定理补充并归纳了其他作者先前获得的结果。