Bessel fractional calculus operators: perspective via Gaussian hypergeometric function

Abstract

This paper investigates the theoretical foundations of Bessel fractional calculus. Distinctions are established between Bessel fractional operators and their classical counterparts. Semigroup theory is developed for the Bessel fractional integral operator, showing that it generates a \(C_{0}\)-semigroup under specific conditions. Fundamental connections between differential and integral operators are analyzed, with comparisons drawn to classical theory. Furthermore, the structure of the solution space for homogeneous fractional differential equations is characterized. The influence of operator parameters–specifically the order and Bessel index \(\nu \) on the resulting integral/differential operations is examined.