发表机构
Ariel University; Poznań University of Technology（阿里埃勒大学; 波兹南理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种有限带方法，用于约化不定积分族，涵盖超指数与代数函数情形，通过构造带状系统实现有效约化，并附有实现软件包。
AI 中文摘要
我们发展了一种用于约化一族不定积分的有限带方法。出发点是对数导数 $K'(x)/K(x)$ 为有理函数的情形。这导致了适配的多项式基和洛朗型基，以及上三角带状矩阵。该框架包含广义实 Schwarz--Christoffel 积分和更一般的超指数权重。然后我们将构造扩展到对数导数 $K'(x)/K(x)$ 在实有理函数域上为代数函数的情形。相关的有限维微分模自然产生块带状系统。最后，我们证明满足具有实多项式系数的线性微分方程的函数会为其矩产生有限递推关系。所得的约化是有效的：对于给定的目标积分，只需解一个有限三角带状系统，并且仅需要与目标相关的约化数据。我们还讨论了算术复杂度，并给出了说明标量和块约化的显式例子。这些算法已在随附的 Mathematica 软件包 \texttt{BandedReduction} 中实现。
英文摘要
We develop a finite-band method for the reduction of families of indefinite integrals. The starting point is the case in which the logarithmic derivative $K'(x)/K(x)$ is a rational function. This leads to adapted polynomial and Laurent-type bases and to upper triangular band matrices. The framework includes generalized real Schwarz--Christoffel integrals and more general hyperexponential weights. We then extend the construction to the case in which the logarithmic derivative $K'(x)/K(x)$ is algebraic over the field of real rational functions. The associated finite-dimensional differential module gives rise naturally to block-band systems. Finally, we show that functions satisfying linear differential equations with real polynomial coefficients give rise to finite recurrences for their moments. The resulting reduction is effective: for a prescribed target integral it is enough to solve a finite triangular band system, and only the reduction data associated with the target are required. We also discuss arithmetic complexity and give explicit examples illustrating the scalar and block reductions. The algorithms are implemented in the accompanying Mathematica package \texttt{BandedReduction}.
