  In this paper, we first propose an unconditionally stable implicit difference
scheme for solving generalized time-space fractional diffusion equations
(GTSFDEs) with variable coefficients. The numerical scheme utilizes the
$L1$-type formula for the generalized Caputo fractional derivative in time
discretization and the second-order weighted and shifted Gr\"{u}nwald
difference (WSGD) formula in spatial discretization, respectively. Theoretical
results and numerical tests are conducted to verify the $(2 - \gamma)$-order
and 2-order of temporal and spatial convergence with $\gamma\in(0,1)$ the order
of Caputo fractional derivative, respectively. The fast sum-of-exponential
approximation of the generalized Caputo fractional derivative and Toeplitz-like
coefficient matrices are also developed to accelerate the proposed implicit
difference scheme. Numerical experiments show the effectiveness of the proposed
numerical scheme and its good potential for large-scale simulation of GTSFDEs.
