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 that often lead to new and sometimes unexpected results. The objects to be studied in this proposal include prime numbers, integer factors of special forms, polynomials, sequences of special matrices and determinants, infinite series, and various special sequences of numbers and functions. Some of the expected results have applications in areas such as graph theory and combinatorics, including lattice paths and integer partitions. These are areas that, in turn, have numerous applications in other sciences, most notably in computer science. Other expected results may not have 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 and coding theory. In fact, my training plan includes supervising graduate theses and undergraduate projects in these last two areas. 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.