Construction of Generalized k-Bessel–Maitland Function with Its Certain Properties

Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O. Box: 1664, Al Khobar 31952, Saudi Arabia Universite de Sousse, Institut Superieur d Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Mathematics, G. F. College, Shahjahanpur 242001, India Department of Mathematics, Integral University Campus, Shahjahanpur 242001, India Department of Mathematics, Aden University, Aden, Yemen


Introduction and Preliminaries
e computation of fragmentary integrals of special functions is significant from the mark of perspective on the value of these outcomes in the assessment of generalized integrals, and the solution of differential and integral equations. Fractional integral formulas involving the Bessel function have been created and assume a significant part in a few physical problems.
e Bessel function is significant in examining the solutions of differential equations, and they are related to a wide scope of problems in numerous regions of mathematical physics, likewise radiophysics, fluid dynamics, and material sciences. ese contemplations have driven different specialists in the field of special functions to investigating the possible expansions and also applications for the Bessel function. Valuable speculation of the Bessel function called the k-Bessel function has also been presented by Diaz et al. [1][2][3] and Suthar et al. [4]. ey have presented k-beta, k-gamma, k-zeta functions, and Pochhammer k-symbol (rising factorial). Additionally, they demonstrated some of their properties and inequalities for the above-said functions. ey have likewise considered k-hypergeometric functions based on k-rising factorial.
Such functions play a discernible role in a variety of appropriate fields of science and engineering. During the past several years, several researchers have obtained various k-type function (such as k-gamma, k-beta, and k-Pochhammer). is subject has received attention of various researchers and mathematicians during the last few decades. e k symbols are well known from many references related to finite difference calculus (see, [5][6][7][8][9][10][11], see additionally [12][13][14][15][16]). Recently, k-type functions and k-type operators have been considered in the literature by various authors. For this purpose, we start with the following properties in the literature.
For our current assessment, we survey here the definition of some known functions and their generalizations. e integral representations of k-gamma and k-beta functions are as follows (see [1][2][3]): where e variety of the functions likewise k-Zeta function, k-Mittag-Leffler function for two and three parameters, k-Wright, and k-hypergeometric functions could be characterized by the following formulas (see also [4,12,13,[16][17][18][19][20]): Definition 1. Let f be a sufficiently well-behaved function with support in R + and let α be a real number α > 0. e k-Riemann-Liouville fractional integral of order α, I α + f is given by (see [21][22][23]) is definition unmistakably reduces the definition defined by Mubeen and Habibullah (see [14]): It is clear that the case k � 1 of (6) yields the traditional Riemann-Liouville fractional integral: Definition 2. Let β be a real number. en, k-Riemann-Liouville fractional derivative is defined by (see [21][22][23]) where Definition 3. For u ∈ φ(R), the fractional Fourier transform (FFT) of order α is defined as (see [21][22][23]) It is effectively observed that, for α � 1, (10) reduces at the conventionally Fourier transform which is given by For w > 0, (10) easily recovers the FFT presented by Luchko et al. [24].
In 2018, Ghayasuddin and Khan [25] presented generalized Bessel-Maitland functions by 2 Journal of Mathematics where For b j , j � 1, q different from nonpositive integers, the series (see [26,27]) is the generalized hypergeometric series, where the Pochhammer symbol and by convention (a) 0 � 1. When p ≤ q, the generalized hypergeometric function converges for all complex values of z, that is, p F q [z] is an entire function. When p > q + 1, the series converges only for z � 0, unless it terminates (as when one of the parameters a j , j � 1, p is a negative integer) in which case it is just a polynomial in z. When p � q + 1, the series converges in the open unit disk |z| < 1 and also for |z| � 1 provided that e summed up k-Wright function is addressed as follows (see details [7,27]): where k ∈ R + ; z ∈ C and Motivated essentially by the demonstrated potential for applications of these extended generalized k-Wright hypergeometric functions, we extend the generalized k-Bessel-Maitland function (18) by means of the generalized k-Pochhammer symbol (1) and investigate certain basic properties including differentiation formulas, integral representations, Euler-Beta, Laplace, Whittaker, and fractional Fourier transforms with their several special cases and relations with the k-Bessel-Maitland function. We also derive the k-fractional integration and differentiation of k-Bessel-Maitland function.

Remark 1.
We note that the case k � 1 in (18) leads to the generalized Bessel-Maitland function defined by Ghayasuddin and Khan [25], which further for δ � p � 1 gives the Bessel-Maitland function given by Singh et al. [20].
In view of Γ k (z + k) � zΓ k (z), we acquire at our stated result (19).
Using Definition 3 on the L.H.S of (20), we get J μ,c+k,δ+k Now, by using the result given in [6], we get J μ,c+k,δ+k

Journal of Mathematics
Using the result (see [6]), we get J μ,c+k,δ+k which is our stated result (20). Proof. With the help of (18) on the L.H.S of (25), we get which is our stated result (25 which is our stated result (26).

Integral Transform of a Generalized k-Bessel-Maitland Function
is section manages with some integral transforms likewise Laplace transform, Whittaker transform, beta transform, Hankel transform, K-transform, and fractional Fourier transform as follows.
In the event that we set the transformation w � s − t/x − t on the L.H.S of equation (30) and using Definition 3, we acquire which is our stated result (30).
Proof. Applying Definition 3, we have By using the following integral (given in [13]) in the above equation, we arrive at (43) In view of (16), we get our stated result (40).
p, q > 0; and q < R(α) + p, then we have Proof. Applying Definition 3 on the L.H.S of (44) and by setting st � z, we get □ Theorem 10 (fractional Fourier transform). e FFT of the generalized k-Bessel-Maitland function for t < 0 is given by (57) On changing variables iw 1/α z � − t and iw 1/α dz � − dt, we arrive at which is our stated result (56).

K-Fractional Integration and K-Fractional Differentiation
Recently, k-fractional calculus gained more attention due to its wide variety of applications in various fields [14,17]. e k-fractional calculus of various types of special functions is used in many research papers [4,28]. For more details about the recent works in the field of dynamic system theory, stochastic systems, non-equilibrium statistical mechanics, and quantum mechanics, we refer the interesting readers to [9,17,24]. In this section, we deduce the outcomes for k-fractional integration and k-fractional differentiation of the above-said function in an orderly way.