Abstract
We investigate two types of dual identities for Riemann fractional sums and differences. The first type relates nabla- and delta-type fractional sums and differences. The second type represented by the Q-operator relates left and right fractional sums and differences. These dual identities insist that in the definition of right fractional differences, we have to use both nabla and delta operators. The solution representation for a higher-order Riemann fractional difference equation is obtained as well.
Highlights
During the last two decades, due to its widespread applications in different fields of science and engineering, fractional calculus has attracted the attention of many researchers [ – ].Starting from the idea of discretizing the Cauchy integral formula, Miller and Ross [ ] and Gray and Zhang [ ] obtained discrete versions of left-type fractional integrals and derivatives, called fractional sums and differences
Several authors started to deal with discrete fractional calculus [ – ], benefiting from the theory of time scales originated by Hilger in
We summarize some of the results mentioned in the above references and add more in the right-type and higher order fractional cases
Summary
During the last two decades, due to its widespread applications in different fields of science and engineering, fractional calculus has attracted the attention of many researchers [ – ].Starting from the idea of discretizing the Cauchy integral formula, Miller and Ross [ ] and Gray and Zhang [ ] obtained discrete versions of left-type fractional integrals and derivatives, called fractional sums and differences. The third section contains some dual identities relating nabla and delta fractional sums and differences in the left and right cases. S=a (ii) The (delta) right fractional sum of order α > (ending at b) is defined by b –αf (t) = (iii) The (nabla) left fractional sum of order α > (starting from a) is defined by
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