Generalized Complex Conformable Derivative and Integral Bases in Fréchet Spaces

This paper presents an additional approach in the field of polynomial bases, utilizing generalized complex conformable fractional derivative and integral operators. These operators are applied to polynomial bases of complex conformable derivatives (GCCDB) and generalized complex conformable integrals (GCCIB) in Fréchet spaces. We also investigate their convergence properties within closed disks, open disks, open regions surrounding closed disks, origin and for all entire functions, employing the Cannon sum, order, type and -property as convergence criteria for our study. The significance of this work lies in generalizing


INTRODUCTION
Theory of basic sets of polynomials (BPs) plays an important role in different mathematical branches.It aids mathematicians and those involved in mathematical fields simplifying studies in areas such as partial differential equations, mathematical physics, and nonlinear analysis.In century mathematicians developed the notion of BPs ( [1], see also [2][3][4]).Their notion depends explicity on an analytic function which can be expanded by the BP { } as ∑ .Taylor series represents the first appearance of the BPs.It was later generalized and expanded to include polynomials such as Lagendre, Euler, Bessel, Bernoulli, Chebyshev polynomials ( [5][6][7][8][9][10]).
One of the crucial issues of the present moment is the fraction calculus (that is, calculus of integrals and derivatives of arbitrary order).Fractional calculus has started to be gained significant attention in a few fields of science and engineering.It has been applied to breakdown numerous dynamical processes and complex nonlinear physical phenomena in physics, electromagnetic, engineering, anomalous diffusion, chemistry, visco-elasticity, and electro-chemistry.Due to its extensive use in the aforementioned domains, this topic has grown in relevance during the past few decades.Numerous recent efforts have been made in this area, some of which are described in [25][26][27].
This paper introduces two new bases, denotes (GCCDB and GCCIB).It investigate the convergence properties of GCCDB and GCCIB in the context of Fréchet spaces, employing tools such as the Cannon function, order, type and -property.Further-more, the paper includes examples and applications related to GCCDB and GCCIB.

Basic concepts
This section presents previous studies on basic sets, Cannon function, order, type and the -property.Additionally, it introduces conceptes related to complex conformable derivatives.
Definition 2.1.The mapping where is a vector space and is the set of real numbers, is a seminorm if the following axioms are satisfied: Given an open disk , with it's closure ̅ , (any open disk enclosing the closed disk ̅ ) and ̅  (some disk surrounding 0).Then, we consider is an entire function ̅ } Then the Fréchet spaces of the sets which mentioned above are given respectively by: Proof.The matrix of coefficients (resp., operators) of { } is given by Using Theorem 2.1, we conclude that { } is also a base.
Similarly, one can deduce that { } is a base, as { } is a base.

Effectiveness of and
This section discusses the effectiveness of { } and { } in Fréchet spaces.This example shows that the two bases have not the same -property.
In [8], the authors establish that Bernoulli polynomials { } are of order 1 and type .Furthermore, Euler polynomial { } are order 1 and type .As an Application of Theorem 5.1, we obtain the following: (i) The GCCD and GCCI of Bernoulli ({ } and { }) are of order 1 and type .
(ii) The GCCD and GCCI of Euler ({ } and { }) are of order 1 and type .

Conclusion
In this study, we have established new polynomial bases in Fréchet spaces by employing generalized complex conformable derivatives and integrable mappings.These generated bases are regraded as a generalization of the work in [29].Notably, when we set , the previous studies become special cases of our work.We also have convergence criteria such as effectiveness, order, type and -property are used for our study.Additionaly, this paper includes applications of Bessel, Bernoulli, Euler and Chebyshev polynomials are added to the paper.
It's noteworthy to say that their study contains variables of domains such as closed hyperball, open hyperball, open hyperball containing closed ball, origin, whole space in Clifford analysis.Another direction of study was pursued by Malonek ([17]), Kishka et al. ([14]), El-sayed ([11]), Kumuyi and Nassif ([15]), El-sayed and Kishka([12]), their researches explored domains such as polycylindrical, hyperspherical, and hyperelliptical regions acrossmultiple complex variables.Hassan et al. ([20]) give a new perspective of BPs of special monogenic polynomials in Fréchet spaces.Because of the importance of the derivative of BPs (DBPs) in topology and algebra, mathematicians have been studied in different regions and spaces.DBPs one