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: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=649294
• 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总体风格可以被描述为“经典”，涉及数十年来人们一直感兴趣的数学对象。然而，新的方法使得重新审视旧物体是值得的，并希望获得新的结果。此外，计算机技术和算法的进步使得进行数学实验（有时是大规模的）成为可能，并将结果用作理论研究的基础，这通常会带来新的、有时是意想不到的结果。******该提案中研究的对象包括素数、特殊形式的非常大整数的因子、多项式、包括多个 zeta 函数的无限级数以及各种特殊的数字和函数序列。一些预期结果在图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直接应用。然而，正如处理大整数、素数或多项式时的通常情况一样，在密码学中总有应用的可能性。虽然这不是拟议研究的主要目的，但我会记住这种可能的应用。我过去和拟议的大部分工作已经（或将）与不同的合著者合作，包括我监督下的学生。******更具体地说，我的短期和中期目标包括以下主题，其中许多工作已经在进行中：*****- Tornheim zeta 函数的解析延拓新方法最近成功地研究了该函数。许多问题仍然存在，这个双级数以及扩展和概括将被探索。***-关于递推关系和伯努利显式扩展以及相关数和多项式的经典结果将被重新审视、统一和扩展。这包括概率论中的最新方法。***- 高斯阶乘（类阶乘乘积）的研究导致了二项式系数的经典同余的有趣扩展和概括；许多问题仍然存在，并将被研究。***- Stern 序列和 Stern-Brocot 树的多项式类似物以及类似的结构对超二元和相关展开、格路径和其他组合问题具有有趣的结果。将探讨这些后果。***- 将检查某些组合和的整除性和同余性。这也与超同余的概念有关。***- 将研究多项式零点的各个方面和应用；该主题还涉及大多数其他主题。*****此外，我将密切关注属于我感兴趣和专业领域的主题的其他问题、项目和可能的合作。