Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case ∗ M. Ferreira †,‡ , N. Vieira ‡ † School of Technology and Management, Polytechnic Institute of Leiria P-2411-901, Leiria, Portugal. E-mail: milton.ferreira@ipleiria.pt ‡ CIDMA - Center for Research and Development in Mathematics and Applications Department of Mathematics, University of Aveiro Campus Universit´ ario de Santiago, 3810-193 Aveiro, Portugal. E-mail: mferreira@ua.pt,nloureirovieira@gmail.com Abstract In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator Δ (α,β,γ) + := D 1+α x + 0 + D 1+β y + 0 + D 1+γ z + 0 , where (α,β,γ) ∈ ]0, 1] 3 , and the fractional derivatives D 1+α x + 0 , D 1+β y + 0 , D 1+γ z + 0 are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator Δ (α,β,γ) + in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions. Keywords: Fractional partial differential equations; Fractional Laplace and Dirac operators; Riemann- Liouville derivatives and integrals of fractional order; Eigenfunctions and fundamental solution; Laplace transform; Mittag-Leffler function. MSC 2010: 35R11; 30G35; 26A33; 35P10; 35A22; 35A08. 1 Introduction In the last decades the interest in fractional calculus increased substantially. This fact is due to on the one hand different problems can be considered in the framework of fractional derivatives like, for example, in optics and quantum mechanics, and on the other hand fractional calculus gives us a new degree of freedom which can be used for more complete characterization of an object or as an additional encoding parameter. For more details about fractional partial differential equations, their applications and their numerical solutions see [8] and the references indicated there. The problems with the fractional Laplace attracted in the last years a lot of attention, due especially to their large range of applications. The fractional Laplace appears e.g. in probabilistic framework as well as in mathematical finance as infinitesimal generators of the stable L´ evy processes [1]. One can find problems involving * The final version is published in Complex Analysis and Operator Theory, 10-No.5, (2016), 1081-1100. It as available via the website http://link.springer.com/article/10.1007/s11785-015-0529-9 1