Discrete-time general fractional calculus

In general fractional calculus (GFC), the counterpart of the fractional time derivative is a differential-convolution operator whose integral kernel satisfies some additional conditions, under which the Cauchy problem for the corresponding time-fractional equation is not only well-posed, but has properties similar to those of classical evolution equations of mathematical physics. In this work, we develop the GFC approach for the discrete-time fractional calculus. In particular, we define within GFC the appropriate resolvent families and use them to solve the discrete-time Cauchy problem with an appropriate analog of the Caputo fractional derivative.