格点行走面积组合学、若干显著三角和与Apéry类数
Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers
AI总结：
本文为推广方格行走及三角格手性行走推导出显式代数面积枚举公式，核心在于将若干显著三角和恒等式重写为多重二项式和，并揭示其与Apéry类数及数论的联系。
摘要：
针对推广标准方格行走的各种格点行走，特别是作者最近引入的三角格手性行走，推导出了显式的代数面积枚举公式。枚举中的一个关键要素是推导出若干涉及三角和的显著恒等式——这些三角和也是诸如Hofstadter模型等重要量子模型的基本构件——并将它们显式重写为多重二项式和的形式。本文还揭示了与数论以及若干类Apéry类数之间的有趣联系，这些Apéry类数是Apéry数的近亲，而Apéry数在ζ(2)和ζ(3)的无理性考虑中起着核心作用。
英文摘要：
Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums --which are also important building blocks of non trivial quantum models such as the Hofstadter model-- and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Apéry-like numbers, the cousins of the Apéry numbers which play a central role in irrationality considerations for ζ(2) and ζ(3).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
