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The main goal in this paper is to study asymptotic behaviour in$L^p(\mathbb{R}^N)$ for the solutions of the fractional version of the discretein time $N$-dimensional diffusion equation, which involves the Caputofractional $h$-difference operator. The techniques to prove the results arebased in new subordination formulas involving the discrete in time Gaussiankernel, and which are defined via an analogue in discrete time setting of thescaled Wright functions. Moreover, we get an equivalent representation of thatsubordination formula by Fox H-functions.

Subordination principle, Wright functions and large-time behaviour for the discrete in time fractional diffusion equation