Sequences and special functions in number theory and combinatorics

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.