Sequences and special functions in number theory and combinatorics

我的许多研究及其总体风格都可以被描述为“经典的”，涉及几十年来经常令人感兴趣的数学对象。然而，新的方法使人们有必要重新审视旧物体，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，这些结果往往会导致新的、有时是意想不到的结果。*本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函数的无穷级数，以及各种特殊的数字和函数序列。一些预期结果在图论或格路等领域有应用，而大多数其他预期结果在数学之外没有直接应用。然而，与通常处理大整数、素数或多项式的情况一样，在密码学中总是有应用的可能性。虽然这不是拟议研究的主要目的，但我会考虑到这些可能的应用。我过去和提议的大部分工作都是与不同的合著者(包括我指导下的学生)共同完成的。*更具体地说，我的中短期目标包括以下主题，其中许多已经在进行中：*-最近成功地研究了Tornheim Zeta函数的解析延拓。仍然有许多问题，将探索这个重级数，以及推广和推广。*-关于伯努利和相关数字和多项式的递归关系和显式展开的经典结果将被重新访问、统一和推广。这包括概率论的最新方法。*-对高斯阶乘(阶乘类乘积)的研究已经导致了经典的二项式系数同余的有趣的扩展和推广；许多问题仍然存在，并将被研究。*-斯特恩序列和斯特恩-布罗科特树的多项式类似和类似的结构对超二进制和相关的展开、格路和其他组合问题具有有趣的结果。我们将探讨这些结果。*-将研究某些组合和的可除性和同余性质。这也与超同余的概念有关。*-将学习多项式零点的各个方面和应用；这个主题也涉及到大多数其他主题。*此外，我还将密切关注其他问题、项目和在我感兴趣和专业领域内的主题上可能的合作。

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.