LAUSR

In this paper we study the fractional analogous of the Laplace–Beltrami equation and the hyperbolic Riesz system studied previously by H. Leutwiler, in $${\mathbb {R}}^3$$R3. In both cases we replace the integer derivatives by Caputo fractional derivatives of order $$0<\alpha \le 1$$0<α≤1. We characterize the space of solutions of the fractional Laplace–Beltrami equation, and we calculate its dimension. We establish relations between the solutions of the fractional Laplace–Beltrami equation and the solutions of the hyperbolic fractional Riesz system. Some examples of the polynomial solutions will be presented. Moreover, the behaviour of the obtained results when $$\alpha =1$$α=1 is presented, and a final remark about the consideration of Riemann–Liouville fractional derivatives instead of Caputo fractional derivatives is made.