1 Introduction and Preliminaries
Fractional calculus is a very rapidly growing subject of mathematics which deals with the study of fractional order derivatives and integrals. Fractional calculus is an efficient tool to study many complex real world systems . It is demonstrated that fractional order representation of complex processes appearing in various fields of science, engineering and finance, provides a more realistic approach with memory effects to study these problems. (see e.g. [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and ). Among the research work developing the theory of fractional calculus and presenting some applications, we have to point out some literature (see [16, 17, 18, 19, 20]). Kumar et al.  analyzed the fractional model of modified Kawahara equation by using newly introduced Caputo-Fabrizio fractional derivative. One also et al.  studied a heat transfer problem and presented a new non-integer model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo-Fabrizio fractional derivative. Recently, one et al.  presented a new fractional extension of regularized long wave equation by using Atangana-Baleano fractional operator. In et al.  one introduced a new numerical scheme for fractional Fitzhugh-Nagumo equation arising in transmission of new impulses. In et al.  one constituted a modified numerical scheme to study fractional model of Lienard’s equations. Hajipour et al.  in their work formulated a new scheme for class of fractional chaotic systems. Baleanu et al.  proposed a new formulation of the fractional control problems involving Mittag-Leffler non-singular kernel. In another work, Baleanu et al.  studied the motion of a Bead sliding on a wire in fractional analysis. Jajarmi et al.  analyzed a hyperchaotic financial system and its chaos control and synchronization by using fractional calculus.
For mathematical modeling of many complex problems appearing in various fields of science and engineering such as fluid dynamics, plasma physics, astrophysics, image processing, stochastic dynamical system, controlled thermonuclear fusion, nonlinear control theory, nonlinear biological systems, quantum physics and heat transfer problems, the fractional calculus operators involving various special functions have been used successfully. There is rich literature available revealing the notable development in fractional order derivatives and integrals (see, [1, 10, 11, 18, 19, 20, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]). Recently, Caputo and Fabrizio  introduced a new fractional derivative which is more suitable than the classical Caputo fractional derivative for many engineering and thermodynamical processes. Atangana  used a new fractional derivative to study the nature of Fisher’s reaction diffusion equation. Riemann and Caputo fractional derivative operators both have a singular kernel which cannot exactly represent the complete memory effect of the system. To overcome these limitations of the old derivatives, very recently Atangana and Baleanu  presented a new non-integer order derivative having a non-local, non-singular and Mittag-Leffler type kernel.
In recent years, many researchers have extensively studied the properties, applications and extensions of various fractional integral and differential operators involving the various special functions. (for detail see McBride , Kalla [44, 45], Kalla and Saxena [46, 47], Saigo [48, 49, 50], Saigo and Maeda , Kiryakova [32, 52],  etc).
For our present study, we recall the following pair of Saigo hypergeometric fractional integral operators.
The following lemmas proved in Kilbas and Sebastin  are useful to prove our main results.
The image formulas for special functions of one or more variables are very useful in the evaluation and solution of differential and integral equations. Motivating by the above discussion, we developed new fractional calculus formulas involving extended generalized Mathieu series.
For various investigations involving the Hadamard product (or the convolution), the interested reader may refer to several recent papers on the subject (see, for example, [69, 70] and the references cited therein).
2 Fractional integration
In this section, we will establish some fractional integral formulas for the generalized (p, q)-Mathieu-type power series. Then their special cases also introduced here.
Proof is parallel to Theorem 1.
2.1 Special cases of fractional integral formulae
In this section we reduces our main findings to the special cases by assigning particular values to the parameters as follows:
3 Image Formulas Associated With Integral Transform
In this section, we establish certain theorems involving the results obtained in previous section associated with the integral transforms like, Beta transform, Laplace transform and Whittaker transform.
3.1 Beta Transform
The proof of this theorem is the same as that of Theorem 3.
3.2 Laplace Transform
The proof of this theorem would run parallel as those of Theorem 5.
3.3 Whittaker Transform
The proof of this theorem would run parallel as those of Theorem 7.
4 Fractional Kinetic Equations
The importance of fractional differential equations in the field of applied science has gained more attention not only in mathematics but also in physics, dynamical systems, control systems and engineering, to create the mathematical model of many physical phenomena. Especially, the kinetic equations describe the continuity of motion of substance. The extension and generalization of fractional kinetic equations involving many fractional operators were found in [72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85].
In view of the effectiveness and a great importance of the kinetic equation in certain astrophysical problems the authors develop a further generalized form of the fractional kinetic equation involving generalized k-Mittag-Leffler function.
5 Solution of generalized fractional kinetic equations
In this section, we investigated the solutions of the generalized fractional kinetic equations by considering generalized (p, q)-Mathieu Type Series
The proof of the Theorem 10 and Theorem 11 are same as that of Theorem 9, so we would like to skip here.
5.1 Special cases
Here we introduce some special cases of our results established in this section.
If p = q, then Theorem 9, Theorem 10 and Theorem 11 reduces to
If p = q = 0, then Theorem 9, Theorem 10 and Theorem 11 reduces to
In the present work, fractional integral formulae involving (p, q)-Mathieu Type series has established. The image formulae of our findings by employing integral transform has been also introduced. Further in this work we gave the solution of fractional kinetic equation in terms of Mittag-Leffler function. All the results are general in nature and give numerous results as their special cases.
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Z. Hammouch, T. Mekkaoui, P. Agarwal, Optical solitons for the calogero-bogoyavlenskii-schiff equation in (2 + 1) dimensions with time-fractional conformable derivative, The European Physical Journal Plus 133:248. doi:)| false https://doi.org/10.1140/epjp/i2018-12096-8. 10.1140/epjp/i2018-12096-8
M. Chand, Z. Hammouch, J. K. Asamoah, D. Baleanu, Certain fractional integrals and solutions of fractional kinetic equations involving the product of s-function, In: Ta? K., Baleanu D., Machado J. (eds) Mathematical Methods in Engineering. Nonlinear Systems and Complexity 24 (2019) 213–244. doi:https://doi.org/10.1007/978-3-319-90972-1_14.
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M. Chand, Z. Hammouch, J. K. Asamoah, D. Baleanu, Certain fractional integrals and solutions of fractional kinetic equations involving the product of s-function, In: Ta? K., Baleanu D., Machado J. (eds) Mathematical Methods in Engineering. Nonlinear Systems and Complexity 24 (2019) 213–244. doi:)| false https://doi.org/10.1007/978-3-319-90972-1_14. 10.1007/978-3-319-90972-1_14
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