Generalized exponential sums and the power of computers

Abstract

Today’s era can be characterized by the rise of computer technology. Computers have been, to some extent, responsible for the explosion of the scientific knowledge that we have today. In mathematics, for instance, we have the four color theorem, which is regarded as the first celebrated result to be proved with the assistance of computers. In this article we generalize some fascinating binomial sums that arise in the study of Boolean functions. We study these generalizations from the point of view of integer sequences and bring them to the current computer age of mathematics. The asymptotic behavior of these generalizations is calculated. In particular, we show that a previously known constant that appears in the study of exponential sums of symmetric Boolean functions is universal in the sense that it also emerges in the asymptotic behavior of all of the sequences considered in this work. Finally, in the last section, we use the power of computers and some remarkable algorithms to show that these generalizations are holonomic; i.e., they satisfy homogeneous linear recurrences with polynomial coefficients.