Exponential laws for fractional differences

In Math. Comp., v. 28, 1974, pp. 185-202, Diaz and Osler gave the following (formal) definition for ${\dot \Delta ^\alpha }f(z)$, the $\alpha$th fractional difference of $f(z):{\dot \Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + \alpha - p)$. They derived formulas and applications involving this difference. They asked whether their differences satisfied an exponent law and what the relation was between their differences and others, such as ${\Delta ^\alpha }f(z) = \Sigma _{p = 0}^\infty A_p^{ - \alpha - 1}f(z + p)$. In this paper an exponent law for their differences is established and a relation found between the two differences mentioned above. Applications of these results are given.