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Exponential functions of discrete fractional calculus

Acar Nihan (Western Kentucky University, Department of Mathematics, Bowling Green, Kentucky, USA)
Atici Ferhan M. (Western Kentucky University, Department of Mathematics, Bowling Green, Kentucky, USA)

In this paper, exponential functions of discrete fractional calculus with the nabla operator are studied. We begin with proving some properties of exponential functions along with some relations to the discrete Mittag-Leffler functions. We then study sequential linear difference equations of fractional order with constant coefficients. A corresponding characteristic equation is defined and considered in two cases where characteristic real roots are same or distinct. We define a generalized Casoratian for a set of discrete functions. As a consequence, for solutions of sequential linear difference equations, their nonzero Casoratian ensures their linear independence.

Keywords: discrete fractional calculus, discrete Mittag-Leffler functions, sequential fractional difference equations