In the present paper, the representation, in different domains, of analytic functions by complex conformable fractional derivative bases (CCFDB) and complex conformable fractional integral bases (CCFIB) in Fréchet space are investigated. Results are proved to show that such representation is possible in closed disks, open disks, open regions surrounding closed disks, at the origin, and for all entire functions. Also, some results concerning the growth order and type of CCFDB and CCFIB are determined. Moreover, the 𝕋�-property of CCFDB and CCFIB is discussed. The obtained results recover some known results when �=1. Finally, some applications to the CCFDB and CCFIB of Bernoulli, Euler, Bessel, and Chebyshev polynomials have been studied.