On the convergence of series of fractional Hasse derivative bases in Fréchet spaces

Abstract

This paper introduces a generalization of the Hasse derivative in the sense of the complex conformable derivative. The definition coincides with the classical version of the Hasse derivative of order . Accordingly, a new base, named complex conformable Hasse derivative bases (CCHDBs), is defined. We investigate the existence of expansions of analytic functions in a series of CCHDBs in Fréchet space on closed and open disks, open regions surrounding closed disks, for all entire functions and at the origin. Moreover, an upper bound for the order and type of the CCHDBs is obtained and proved to be attainable. The ‐property of CCHDBs is also discussed. Our results improve and extend the analog results in the complex analysis related to the classical complex derivative of a base of polynomials (BPs). The obtained results clarify several implications for the CCHDBs of special functions such as Euler, Bernoulli, Chebyshev, Bessel, and Gontcharaff polynomials.