Sequences and special functions in number theory and combinatorics

数论和组合学中的序列和特殊函数

• 批准号: RGPIN-2017-05144
• 负责人: Dilcher, Karl
• 金额: $ 1.75万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710711
• 关键词: Sequences; special; functions; number; theory

Much of my research and its general flavour can be described as "classical", dealing with mathematical objects that have often been of interest for decades. However, new methods make it worthwhile to take a fresh look at old objects, with the hope of obtaining new results. Also, advances in computer technology and in algorithms make it possible to do mathematical experiments, sometimes at a large scale, and use the outcomes as a basis for theoretical investigations which often lead to new and sometimes unexpected results. The objects studied in this proposal include prime numbers, factors of very large integers of special forms, polynomials, infinite series including multiple zeta functions, and various special sequences of numbers and functions. Some of the expected results have applications in areas such as graph theory or lattice paths, while most other expected results have no immediate applications outside of mathematics. However, as is usually the case when one deals with large integers, prime numbers, or polynomials, there is always the possibility of applications in cryptography. Although this is not the main purpose of the proposed research, I will keep such possible applications in mind. Most of my past and proposed work has been (or will be) joint with various co-authors, including students under my supervision. To be more specific, my short- and medium-term goals include the following topics, many of which are already works in progress: - A new approach to the analytic continuation of the Tornheim zeta function has recently been successful in studying this function. Numerous questions remain, and this double series, along with extensions and generalizations, will be explored. - Classical results on recurrence relations and explicit expansions of Bernoulli and related numbers and polynomials will be revisited, unified, and extended. This includes recent methods from the theory of probability. - The study of Gauss factorials (factorial-like products) has led to interesting extensions and generalizations of classical congruences for binomial coefficients; many questions remain, and will be investigated. - Polynomial analogues of the Stern sequence and Stern-Brocot tree and similar structures have interesting consequences to hyperbinary and related expansions, lattice paths, and other combinatorial questions. These consequences will be explored. - Divisibility and congruence properties of certain combinatorial sums will be examined. This is also related to the concept of supercongruences. - Various aspects and applications of zeros of polynomials will be studied; this topic also reaches into most of the other topics. In addition, I will keep my eyes open for other problems, projects, and possible collaborations on topics that fall within my areas of interest and expertise.

我的大部分研究及其一般风味可以描述为“古典”，处理数十年来通常引起的数学对象。但是，新方法值得重新看一下旧物体，并希望获得新的结果。同样，计算机技术和算法的进步使得有时会大规模进行数学实验，并将结果作为理论研究的基础，这通常会导致新的，有时甚至是意外的结果。 该提案中研究的对象包括质数，特殊形式的非常大整数的因素，多项式序列，包括多个ZETA函数以及各种数字和功能的特殊序列。一些预期的结果在图理论或晶格路径等领域有应用，而大多数其他预期结果在数学之外没有立即应用。但是，与通常处理大整数，质数或多项式的情况一样，加密术中总是有应用。尽管这不是拟议研究的主要目的，但我将牢记这种可能的应用程序。我过去和拟议的大部分工作都是（或将）与各种合着者，包括在我的监督下的学生。 更具体地说，我的短期和中期目标包括以下主题，其中许多主题已经在进行中： - 最近在研究此功能方面成功地成功地研究了龙卷风Zeta功能的分析延续的新方法。仍有许多问题，将探讨这个双重系列以及扩展和概括。 - 将重新审视，统一和扩展有关Bernoulli以及相关数字和多项式的显式扩展的经典结果。这包括概率理论的最新方法。 - 对高斯阶乘（类似阶乘的产品）的研究导致了有趣的扩展和经典一致性的二项式系数；仍然存在许多问题，并将进行调查。 - 船尾序列和船尾树木和类似结构的多项式类似物对超循环和相关膨胀，晶格路径和其他组合问题产生了有趣的后果。这些后果将被探讨。 - 将检查某些组合总和的分裂性和一致性。这也与超级企业的概念有关。 - 将研究多项式零的各个方面和应用；该主题也涉及大多数其他主题。 此外，我将睁开眼睛，了解其他问题，项目以及可能属于我感兴趣和专业领域的主题的合作。