Geometry MMA Seminar: How to expand a number or a rational function as a sum of rapidly converging reciprocals
Mar 27, 2023, 10:30 - 11:30 AM
Speaker: Neil Epstein, GMU
Title: How to expand a number or a rational function as a sum of rapidly converging reciprocals
Abstract: How can one represent a proper fraction as a sum (or alternating sum) of distinct unit fractions? How can one represent a real number in the unit interval as a convergent series of unit fractions? It turns out there are infinitely many ways for any given number, but when restrictions are placed, uniqueness may be obtained. This has been known for over a century, but seems not to be widely known in the mathematical community, so some of my talk will explain some of the more popular ways. I will focus on those that can be seen as a positive-slope version of simple continued fractions, most especially the Engel and Pierce expansions, where each successive denominator is a multiple of the previous ones.
One can ask an analogous question of rational functions. Given a proper rational function (i.e. deg(numerator) < deg(denominator)), is there an algorithm to represent it as something analogous to a Pierce or Engel expansion — i.e. a sum or alternating sum of reciprocals of polynomials, where each successive polynomial is a polynomial multiple of the previous one? I have proved that the answer is “yes”, and that the uniqueness properties are somewhat stronger than those for Engel and Pierce expansions of a rational number.
Time: Monday, March 27 – 10:30am-11:30am
Place: Exploratory Hall, Room 4106