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.

我的大部分研究和它的一般味道可以被描述为“经典”，涉及几十年来经常感兴趣的数学对象。然而，新的方法使我们有必要重新审视旧的对象，以期获得新的结果。此外，计算机技术和算法的进步使得数学实验成为可能，有时是大规模的，并将结果作为理论研究的基础，这往往会导致新的，有时是意想不到的结果。在这个建议中研究的对象包括素数，特殊形式的非常大的整数的因子，多项式，无穷级数，包括多个zeta函数，以及各种特殊的数字和函数序列。一些预期的结果在图论或格路径等领域有应用，而大多数其他预期的结果在数学之外没有直接的应用。然而，通常情况下，当一个人处理大整数，素数，或多项式，总是有应用在密码学的可能性。虽然这不是拟议研究的主要目的，但我会记住这些可能的应用。我过去和计划中的大部分工作都是（或将是）与各种合著者联合进行的，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，其中许多已经在进行中：**-Tornheim zeta函数解析延拓的新方法最近成功地研究了该函数。许多问题仍然存在，这个双重系列，沿着扩展和概括，将被探索。经典的结果递归关系和显式展开的伯努利和相关的数字和多项式将被重新审视，统一，并扩大。这包括最近的概率论方法。对高斯因子（类因子积）的研究导致了对二项式系数的经典同余的有趣的扩展和推广;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进制和相关的展开式、格路径和其他组合问题有着有趣的影响。这些后果将予以探讨。*某些组合和的整除性和同余性质将被检查。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;本主题还涉及大多数其他主题。**此外，我将继续关注其他问题，项目，以及在我感兴趣和专业领域内的可能合作。