A new fractional difference with an exponential kernel function is proposed in this study. First, a difference operator is defined by the exponential function. From the Cauchy problem of the nth-order difference equation, new fractional-order sum and differences are presented. The propositions between each other and numerical schemes are derived. Finally, fractional linear difference equations are presented, and exact solutions are given by using a new discrete Mittag-Leffler function.
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