Exponential operators and solution of pseudo-classical evolution problems Caterina Cassisa, Paolo E. Ricci, Universit` a di Roma “La Sapienza”, Dipartimento di Matematica, P.le A. Moro, 2 00185 Roma, Italia - e-mail: cassisa@uniroma1.it, riccip@uniroma1.it Ilia Tavkhelidze “I.N. Vekua” Institute of Applied Mathematics, Tbilisi State University 2, University Street – 380043 – Tbilisi (Georgia) - e-mail: iliko@viam.hepi.edu.ge Abstract We use the multi-dimensional polynomials considered by Hermite, and subse- quently studied by P. Appell and J. Kamp´ e de F´ eriet, in order to obtain explicit solutions of pseudo-classical PDE problems in the half-plane y> 0. We consider systems of PDE, including some problems with degeneration on the x-axis. 2000 Mathematics Subject Classification. 33C45, 44A45, 35G15. Key words and phrases. Hermite-Kamp´ e de F´ eriet polynomials, Operational calculus, Pseudo-hyperbolic and Pseudo-circular functions, Boundary value problems. 1 Introduction In preceding articles [3], [4], [5], by using an operatorial approach, we showed that the solution of many classical or pseudo-classical boudary value problems in the half plane for PDE, (with constant coefficients and analytic boundary, or initial, data) can be expressed in terms of the higher order Hermite-Kamp´ e de F´ eriet (shortly H-KdF) polynomials. The relevant results were extended to the multi-dimensional case in [13], [14]. In this article we extend the results of our article [5] to the pseudo-classical case too. We start recalling properties of the H-KdF polynomials and extending technical tools considered in [3], [4] to the case under examination.