论文标题
简而
Ansatz in a Nutshell: A comprehensive step-by-step guide to polynomial, $C$-finite, holonomic, and $C^2$-finite sequences
论文作者
论文摘要
给定一个序列1、1、5、23、135、925、7285、64755、641075、6993545、83339745，...，我们如何猜出一个公式？本文将迅速介绍Ansatz的概念，用于多项式，$ c $ -finite，载体和最新添加$ c^2 $ -Finite序列。对于这些类别，我们详细讨论了猜测和检查，生成功能，封闭属性和封闭形式的解决方案的各个方面。每个定理都有一个可访问的证据，其次是旨在激发理论发展的几个例子。每个示例都伴随着枫计划，目的是证明该程序在解决该领域的问题中使用。尽管这项工作旨在对现有的Ansatzes进行全面的审查，但我们还通过为$ C^2 $ -FINITE序列提供理论和数值结果来系统地填补文献中的研究空白。我们希望读者将享受我们统一的框架学习安萨兹的旅程。
Given a sequence 1, 1, 5, 23, 135, 925, 7285, 64755, 641075, 6993545, 83339745,..., how can we guess a formula for it? This article will quickly walk you through the concept of ansatz for classes of polynomial, $C$-finite, holonomic, and the most recent addition $C^2$-finite sequences. For each of these classes, we discuss in detail various aspects of the guess and check, generating functions, closure properties, and closed-form solutions. Every theorem is presented with an accessible proof, followed by several examples intended to motivate the development of the theories. Each example is accompanied by a Maple program with the purpose of demonstrating use of the program in solving problems in this area. While this work aims to give a comprehensive review of existing ansatzes, we also systematically fill a research gap in the literature by providing theoretical and numerical results for the $C^2$-finite sequences. We hope the readers will enjoy the journey through our unifying framework for the study of ansatz.
