Sequences and special functions in number theory and combinatorics

我的许多研究及其总体风格可以被描述为“经典”，涉及几十年来经常引起人们兴趣的数学对象。然而，新的方法使人们有必要重新审视旧的对象，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。******本建议研究的对象包括素数、特殊形式的大整数因子、多项式、包含多个ζ函数的无穷级数以及各种特殊数列和函数。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，在处理大整数、素数或多项式时，通常都有在密码学中应用的可能性。虽然这不是提议的研究的主要目的，但我会记住这些可能的应用。我过去和计划的大部分工作已经（或将）与各种共同作者合作，包括我指导下的学生。******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函数的解析延拓的新方法最近在研究该函数方面取得了成功。许多问题仍然存在，这个双级数，以及扩展和推广，将被探索。***-经典结果递归关系和显式展开伯努利和相关的数字和多项式将被重新审视，统一和扩展。这包括来自概率论的最新方法。***-高斯阶乘（类阶乘积）的研究导致了二项式系数经典同余的有趣扩展和推广；仍有许多问题有待调查。***- Stern序列和Stern- brocot树的多项式类似物以及类似结构对超二进制和相关展开、格路径和其他组合问题具有有趣的影响。我们将探讨这些后果。***-将研究某些组合和的可整除性和同余性。这也与超同余的概念有关。***-将研究多项式零的各个方面和应用；这个话题也涉及到大多数其他话题。******此外，我将继续关注我感兴趣和专业领域内的其他问题、项目和可能的合作。

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.