关于与“分形山”有关的几个显式积分
On some explicit integrals related to "fractal mountains"
摘要：
循环计数函数$U(x)$估计$x\in[-1,1]$的数字表示中的“加权”循环的数目。粗略地说，每个$x$被认为是一个无限行走，其中行走的步骤对应于$x$的数字。循环计数函数$U$的图形具有类似于复杂山地景观的分形结构。在某种意义上，$U$允许我们全局地观察随机游动。由于自回避随机游动与$U(x)$的零点密切相关，这些函数在分析与自回避随机游动（SAW）分布有关的一些困难问题时可能是有用的.这里我们注意到，$U(x)$可以自然地扩展为多维变元$x$。在这篇文章中，重点将放在一些分析方面。它将表明，积分$\int x^AU(x)^Bdx$与非负整数$A$和$B$可以表示为整数系数的有理函数的积分。此外，它将表明，$\int x^A U(x)dx$承认封闭形式的表达式。还计算了$U$的傅立叶级数。最后，我们讨论了与特殊函数和广义连分式的一些联系，以及其他观点。
英文摘要：
Loop counting functions $U(x)$ estimate the number of "weighted" loops in a digital representation of $x\in[-1,1]$. Roughly speaking, each $x$ is considered as an infinite walk, where the steps of the walk correspond to digits of $x$. The graph of loop counting functions $U$ has a fractal structure that resembles complex mountain landscapes. In some sense, $U$ allows us to look at random walks globally. These functions may be helpful in the analysis of some hard problems related to the distribution of self-avoiding random walks (SAW) in a multi-dimensional case since SAW closely relate to zeros of $U(x)$. We note here that $U(x)$ can be naturally extended to a multidimensional argument $x$. In this article, the focus will be on some analytic aspects. It will be shown that integrals $\int x^AU(x)^Bdx$ with non-negative integers $A$ and $B$ can be expressed in terms of integrals of rational functions with integer coefficients. Moreover, it will be shown that $\int x^A U(x)dx$ admits closed-form expressions. Fourier series for $U$ is also computed. Finally, we discuss some connections with special functions and generalized continued fractions, and other perspectives.
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