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Fractional \(q\)-calculus on a time scale. (English) Zbl 1157.81315

Summary: The study of fractional \(q\)-calculus in this paper serves as a bridge between the fractional \(q\)-calculus in the literature and the fractional \(q\)-calculus on a time scale \(\mathbb T_{t_0}= \{t:t=t_0q^n, n \text{ a nonnegative integer }\}\cup\{0\}\), where \(t_0\in\mathbb R\) and \(0<q<1\). By use of time scale calculus notation, we find the proof of many results more straight forward. We develop some properties of fractional \(q\)-calculus, we develop some properties of a \(q\)-Laplace transform, and then we shall employ the \(q\)-Laplace transform to solve fractional \(q\)-difference equations.

MSC:

81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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