Extended Bessel-Maitland function and its properties pertaining to integral transforms and fractional calculus

: The aim of this paper is to establish an (presumably new) extension of generalized Bessel-Maitland function by using the extension of extended beta function. In addition, investigate several important properties namely integral representation, derivatives, recurrence relation, Beta transform and Mellin transform. Further, certain properties of the Riemann-Liouville fractional calculus associated with extended generalized Bessel-Maitland function are also investigated.


Introduction and preliminaries
In applied sciences, many important functions are defined via improper integrals or series (or finite products). The general name of these important functions knows as special functions. In special function, one of the most important function (Bessel function) has gained importance and popularity due to its applications in the problem of cylindrical coordinate system, wave propagation, heat conduction in cylindrical object and static potential etc. In the recent years, some generalizations (unification) and number of integral transforms of Bessel functions have been given by many mathematicians and physicist as well as engineers. The Bessel-Maitland function J τ ϑ (z) is a generalization of Bessel function, defined in [7] through a series representation as: (−z) n Γ (τn + ϑ + 1) n! (1. 1) In fact, the application of Bessel-Maitland function are found in the diverse field of mathematical physics, engineering, biological, chemical in the book of Watson [26].
Further, generalization of the generalized Bessel-Maitland function defined by Pathak [13] is as follow: Motivated by the established potential for application of these Bessel-Maitland functions, we introduce here another interesting extension of the generalized Bessel-Maitland function as follow: which will be known as extended generalized Bessel Maitland function (EGBMF).
Here, B ω (x, y; p) is an extension of extended beta function introduced by Parmar et al. [12] in the following way: where K ω+ 1 2 (.) is the modified Bessel's function. The special case of (1.4) corresponding to ω = 0 be easily seen to reduce to the extended beta function upon making use of ( [9], Eq (10.39.2)). If p = 0 in Eq (1.5), reduces in to the classical beta function. For a detailed account of various properties, generalizations and applications of Bessel-Maitland functions, the readers may refer to the recent work of the researchers [3,15,[21][22][23][24][25] and the references cited therein.
Reciprocate the order of summation and integration, that is surd under the presumption given in the description of Theorem 2.1, we get Using Eq (1.2) in Eq (2.3), we obtain the desired result Eq (2.1).
Corollary 2.2. Let the condition of Theorem 2.1 be satisfied, the following integral representation holds: Proof. By taking t = r 1+r in Theorem 2.1, After simplification, we obtain the desired result Eq (2.4).
Corollary 2.3. Assume the state of Theorem 2.1 is satisfied, the following integral representation holds: Proof. If we set t = sin 2 θ in Theorem 2.1, we acquire the above result.

Recurrence relation
then the recurrence relation holds true: Proof. Employing Eq (1.3) in right hand side of Eq (3.1), we obtain

Derivative formulae
Theorem 4.1. For the extended generalized Bessel Maitland function, we have the following higher derivative formula: Proof. Taking the derivative with respect to z in Eq (2.1), we get Again taking the derivative with respect to z in Eq (6.5), we get Ongoing the repetition of this technique n times, we get the desired result Eq (4.1).
Theorem 4.2. For the extended generalized Bessel Maitland function, the following differentiation holds: Proof. Replace z by σz τ in Eq (2.1) and take its product with z ϑ , then taking z-derivative n times. We obtain our result.

Beta transform
Definition 5.1. The Beta transform [19] of a function f (z) is defined as: (z τ ; p) ; ϑ + 1, 1 Upon interchanging the order of summation and integration in Eq (5.3), which can easily verified by uniform convergence under the constraint with Theorem 5.2, we get Using the familiar definition of beta function, and interpreting with Eq (1.3), we get the desired representation Eq (5.2).
Proof. Using the definition of Melllin transform (6.1) and (1.3), we obtain Interchanging the order of integration in Eq (6.5), which is admittable because of the conditions in the statement of the Theorem 3.4, we get Now taking u = p t(1−t) in Eq (6.6), we get From Olver et al. [9]: Applying Eq (6.8) in Eq (6.7), we obtain Using Eq (1.2), and interchanging the order of summation and integration which is valid for (τ) > 0, (ϑ) > 0, (s) > 0, (s) > (ς) > 0, (s + ξ − ς) > 0, we obtain Using the relation between Beta function and Gamma function, we obtain After simplification, we obtain In view of Eq (6.3), we arrived at our result Eq (6.4).

Fractional calculus approach
In recent years, the fractional calculus has become a significant instrument for the modeling analysis and assumed a significant role in different fields, for example, material science, science, mechanics, power, science, economy and control theory. In addition, research on fractional differential equations (ordinary or partial) and other analogous topics is very active and extensive around the world. One may refer to the books [28,31], and the recent papers [1,2,6,16,18,27,29,30,[32][33][34][35] on the subject. In this portion, we derive a slight interesting properties of EMBMF associated with the right hand sided of Riemann-Liouville (R-L) fractional integral operator I ζ a+ and the right sided of R-L fractional derivative operatorD ζ a+ , which are defined for ζ ∈ C, ( (ζ) > 0), x > 0 (See, for details [5,17]): and where (ζ) is the integral part of (ζ). A generalization of R-L fractional derivative operator (7.2) by introducing a right hand sided R-L fractional derivative operator D ζ, σ a+ of order 0 < ζ < 1 and 0 ≤ σ ≤ 1with respect to x by Hilfer [4] is as follows: The generalization Eq (7.3) yields the R-L fractional derivative operator D ζ a+ when σ = 0 and moreover, in its special case when σ = 1, the definition (7.3) would reduce to the familiar Caputo fractional derivative operator [5].

Conclusions
In the present paper, The properties, integral transform and fractional calculus of the newly defined extended generalized Bessel-Maitland type function are investigated here and find their connection with other functions scattered in the literature of special function. Various special cases of the derived results in the paper can be evaluate by taking suitable values of parameters involved. For example if we set ω = 0, β = β − 1 and z = −z in (1.3), we immediately obtain the result due to Mittal et al [18]. For various other special cases we refer [19,21] and we left results for the interested readers.