The Extended Bessel-Maitland Function and Integral Operators Associated with Fractional Calculus

)e aim of this paper is to introduce a presumably and remarkably altered integral operator involving the extended generalized Bessel-Maitland function. Particular properties are considered for the extended generalized Bessel-Maitland function connected with fractional integral and differential operators. )e integral operator connected with operators of the fractional calculus is also observed. We point out important links to known findings from some individual cases with our key outcomes.


Definition 1.
e space of Lebesgue measurable of real or complex valued function L(a, b) for our study of the significance of fractional calculus is defined as follows: Definition 2. e Riemann-Liouville (R-L) fractional integral operators I ℓ a+ and I ℓ b− are defined respectively as (see, e.g., [13]) follows: (8) where f(x) ∈ L(a, b), ℓ ∈ C, and R(ℓ) > 0.
We found the following baseline findings for our study.

Lemma 2 (Srivastava and Manocha [17]). If a function f(z) is analytic and has a power series representation
Lemma 3 (Srivastava and Tomovski [18]). Let en, the subsequent result holds true for L ℓ,v a+ f as follows: We also provided the subsequent established facts and rules in this article.
Fubini's theorem (Dirichlet formula) (Samko et al. We define the following integral operator in terms of extended generalized Bessel function for δ, ω ∈ C, R(ς) > 0 and R(ϑ) > 0 for our further analysis of fractional calculus, then the integral operator where x > a. If we put p � 0 to the operator, then (15) reduces If ω � 0, then (16) reduces the integral operator to the R-L fractional integral operator described in (7).

Integral Operators with Extended Generalized Bessel-Maitland Function in the Kernel
In this part, we consider the composition of the fractional integral and derivative of Riemann-Liouville and the fractional derivative of Hilfer with the extended generalized Bessel-Maitland function defined by (4).
, and q, n ∈ N, then the following result holds true: Proof. Using (4), we see that Using the identity, and after simplifying, we have d dz Finally, it can be expressed by using (4) again, and we obtain d dz n z ϑ J ς,δ;c ϑ,q ωz ς ; p � z ϑ−n J ς,δ;c ϑ−n,q ωz ς ; p .