On quantum trigonometric fractional calculus

In this research work, we introduce an innovative concept known as the q -trigonometric derivative within the framework of the Caputo derivative. Our approach begins by introducing a novel notion of a trigonometric q -derivative and thoroughly examining its characteristics. We subsequently merge this new definition with the Caputo derivative to introduce a novel approach to q -fractional calculus. To analytically address this q -trigonometric system, we effectively employ the q -Laplace transform to derive solutions. Notably, the bivariate q -Mittag-Leffler ( q -ML) function plays a significant role in this process. We provide detailed explanations and examples of this approach with two illustrative examples.