Generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations associated to the number operator

By means of the Laplace transform, we give the solution of the generalized Riemann-Liouville and Liouville-Caputo time fractional evolution equations in infinite dimensions associated to the number operator. These solutions are given in terms of the Mittag-Leffler function and the convolution product.