(joint work with Duco van Straten)
Power series expansion with meromorphic functions as coefficients are ubiquitous in mathematics. They appear in the work of Euler, Ramanujan as q-analogs of hypergeometric functions, in dynamical systems in the work of Arnold and are widely used in mathematical physics.
In 1912, E. Borel started to develop a theory of monogenic functions to extend the classical theory of holomorphic function. Unlike holomorphic functions, monogenic functions are not uniquely defined by their Taylor expansions. However, Borel found a quasi-analytic class of monogenic functions that is a subspace which has the same properties as the space of holomorphic functions: functions are uniquely defined by their asymptotic expansion at a point. Unfortunately Borel class is very restrictive, it does not contain the most simple examples of q-analogs.
Recently with Duco van Straten, we introduced a quasi-analytic class that we call meandromorphic functions because their zero locus sometimes have the form of a meander. This class is large enough to include q-analogs and classical perturbative expansions of classical and quantum mechanics. This explains in particular why most perturbative series of mathematical physics diverge: meandromorphic normal forms have poles at resonances and therefore these tend to have many accumulation points, a phenomenon clearly impossible for a basic holomorphic function.
Text: arxiv 2410.04583
Vladimir Rubtsov, Vasily Golyshev, Ilia Gaiur
