Iterated Integral Representation of the Characteristic Function of Time-Delay Systems With Spectral Values of Maximal Multiplicity

Motivated by recently obtained factorizations of characteristic functions of some classes of time-delay systems admitting a root with the largest possible multiplicity in terms of Kummer confluent hypergeometric functions, this letter provides a new representation of some Kummer functions with integer coefficients in terms of iterated integrals of an exponential kernel. As a consequence of the existing links between Kummer, Whittaker, and modified Bessel functions, the latter classes of special functions also take advantage of such an iterated integral representation. We also express characteristic functions of the aforementioned classes of time-delay systems in terms of iterated integrals, and illustrate how such an iterated integral representation allow us to obtain information on the location of the spectrum of the system, at least in some low-order cases.