Stieltjes级数收敛因子的阶乘级数表示
Factorial Series Representation of Stieltjes Series Converging Factors
AI总结：
本文提出将任意阶Stieltjes级数收敛因子表示为逆阶乘级数，并通过简单递归算法解析求取各项，为发展Weniger变换在Stieltjes级数上的一般收敛理论奠定初步基础。
摘要：
Levin型非线性序列变换作为数值工具，用于发散级数求和或缓慢收敛级数的收敛加速，其实际效用如今已无可争议。特别是Weniger变换，在解决重求和问题时能够取得引人注目的结果，常常优于更为知名的重求和技术，其中最著名的是Padé逼近。然而，我们对其理论特性的理解仍远不能令人满意，尤其是在阶乘发散级数的解码方面尤为薄弱。Stieltjes级数代表了一类在数学物理中具有根本重要性的幂级数。本文展示了任意阶Stieltjes级数收敛因子如何可表示为一个逆阶乘级数，其各项可通过一个简单的递归算法解析求得。文中给出了若干应用该算法的例子，以展示其有效性和实现上的简便性。我们相信，这里给出的结果可能构成发展Weniger变换在Stieltjes级数上的一般收敛理论的重要初步步骤。这是一个相当宏大的项目，但值得在未来继续推进。
英文摘要：
The practical usefulness of Levin-type nonlinear sequence transformations as numerical tools for the summation of divergent series or for the convergence acceleration of slowly converging series, is nowadays beyond dispute. Weniger's transformation, in particular, is able to accomplish spectacular results when used to overcome resummation problems, often outperforming better known resummation techniques, the most known being Padé approximants. However, our understanding of its theoretical features is still far from being satisfactory and particularly bad as far as the decoding of factorially divergent series is concerned. Stieltjes series represent a class of power series of fundamental interest in mathematical physics. In the present paper, it is shown how the Stieltjes series converging factor of any order is expressible as an inverse factorial series, whose terms can be analytically retrieved through a simple recursive algorithm. A few examples of applications of our algorithm are presented, in order to show its effectiveness and implementation ease. We believe the results presented here could constitute an important, preliminary step for the development of a general convergence theory of Weniger's transformation on Stieltjes series. A rather ambitious project, but worthy of being pursued in the future.
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