Generalized Differential and Integral Operators

Abstract

From the time of discovery of calculus by Leibniz, he studied the problem of fractional differentiations. 30 September 1695, the day when Leibniz sent a letter to L’Hôpital with a reply of the L’Hôpital’s question related to the differentiation of a function of order n = 1∕2, became a birthday of the fractional calculus. By using the Leibniz product rule and the binomial theorem he obtained some paradoxical results. Euler partially resolved the Leibniz paradox by introducing the gamma function as 1 ⋅ 2 ⋅… ⋅ n = n! = Γ(n + 1). Therefore, the fractional calculus has attracted attention to a range of celebrated mathematicians and physicists, such as Leibniz, Euler, Laplace, Lacroix, Fourier, Abel, Liouville, Riemann, Grünwald, Letnikov, to name but a few.