In the present era, fractional calculus plays an important role in various fields. Fractional Calculus is a field of mathematic study that grows out of the traditional definitions of the calculus integral and derivative operators in much the same way fractional exponents is an outgrowth of exponents with integer value. Based on the wide applications in engineering and sciences such as physics, ...

In the geometric function theory of complex analysis, investigation properties analytic functions using q-analogues differential and integral operators is an important area study, offering powerful tools for applications in numerical analysis solution equations. Many topics, including hypergeometric series, particle physics, have been generalized q-calculus. this first all, we define a operator...

Fractional calculus and q-deformed Lie algebras are closely related. Both concepts expand the scope of standard Lie algebras to describe generalized symmetries. For the fractional harmonic oscillator, the corresponding q-number is derived. It is shown, that the resulting energy spectrum is an appropriate tool e.g. to describe the ground state spectra of even-even nuclei. In addition, the equiva...

We prove that the operators in a class of rough fractional integral operators and the related maximal operators are bounded from L p (v p) to L q (u q) with weight pair (u,v).

We conduct a formal study of particular class fractional operators, namely weighted calculus, and its extension to the more general known as calculus with respect functions. emphasise importance conjugation relationships classical Riemann–Liouville use them prove many fundamental properties these operators. As examples, we consider special cases such tempered, Hadamard-type, Erdélyi–Kober also ...

Abstract A generalized q-Taylor’s formula in fractional q-calculus is established and used in deriving certain q-generating functions for the basic hypergeometric functions and basic Fox’s H-function. 2010 Mathematics Subject Classification: 33D45, 26A33 and 33D90.