Kernel families of fractional derivatives

Fibonacci is known in particular due to the notion of the golden ratio, which has found great success in modeling natural phenomena. Also, based on this ratio, the Fibonacci polynomials emerged, which are employed in this work to establish an innovative family of fractional kernels. After showing that the family of kernels is well defined, we define two fractional derivatives, one of Caputo type and one of Riemann–Liouville type. Next, we demonstrate certain bound characteristics that the newly defined derivatives have. We also define the associated fractional integral for one of the family members of the newly defined derivative. Using the method of Laplace transform, we found explicit solutions for $RC$ electrical circuits, using the fractional derivative introduced in this work.