A new approach to multi-delay matrix valued fractional linear differential equations with constant coefficients

In this work we obtain a new generalization of the multi-delay perturbation of the two parameter Mittag-Leffler function introduced recently by Mahmudov in the context of multi-delay fractional differential equations [36]. The generalization is introduced to extend Mahmudov results to include besides discrete delay also pantograph fractional differential equations and can be useful to generalize previous work due to Putzer on the representation of the usual matrix exponential avoiding the Jordan canonical form [46]. We adapt Putzer's representation to the fractional setting with delay using an extension of the Omega Matrix Calculus in order to deal with matrices over a field of fractions of a suitable commutative ring instead of the usual complex field. In contrast to previous representations, ours involves only a fixed finite number of matrix powers and no spectral information is required to compute the solutions of the fractional differential equations. Furthermore, by specializing to integer order systems we show that several results of the literature are special cases of our general formulation including a version of Putzer's original representation. The Dickson polynomials of the second kind play an intrinsic and important role in our new representation.