New concepts of fractional Hahn’s q,ω $q,\omega $-derivative of Riemann–Liouville type and Caputo type and applications

Abstract

Abstract In this paper, we give the definitions of q,ω $q,\omega $-exponential function and q,ω $q,\omega $-gamma function. New concepts of fractional Hahn’s q,ω $q,\omega $-derivative of Riemann–Liouville type and Caputo type are introduced, meanwhile we discuss some properties. As applications of the new concepts, we give the existence result of positive solutions for boundary value problem of fractional q,ω $q,\omega $-derivatives equations. We also definite certain q,ω $q,\omega $-Mittag-Leffler function by solving the initial value problem of fractional q,ω $q,\omega $-equations.

Keywords

• Quantum calculus
• q , ω $q,\omega $ -derivative
• Boundary value problem
• Initial value problem