Rational approximations of functions offer a rich mathematical theory. Touching subjects such as orthogonal polynomials, potential theory and of course differential equations. In this talk we will discuss a specific type of rational approximant, factorial expansions. In recent work with O. Costin and R. Costin we have developed a theory of dyadic expansions which improve the domain and rate of convergence when compared to the classical methods found in the literature. These results provide a general method for producing rational approximations of Borel summable series with locally integrable branch points. Surprisingly, these expansions capture the asymptoticly important Stokes phenomena. Additionally, we find applications in operator theory on Hilbert spaces providing new representations for (bounded and unbounded) positive and self-adjoint operators in terms of the semigroups and unitary groups they generate. Finally, as an example of an important application we discuss representing the tritronquée solutions of Painlevé’s first equation.

A recording of the talk can be found at
https://youtu.be/1KC3Ah1fpFY