Some analytical merits of Kummer-Type function associated with Mittag-Leffler parameters

Abstract As of late, the study of Fractional Calculus (FC) and Special Functions (SFs) has been interestingly prompted in various realms of mathematics, engineering and sciences. This is due to the considerable demonstrated potential of their applications. Among these SFs, Gamma function and Mittag-Leffler functions are the most renowned and distinguished. Numerous authors continue to study this line. The current analysis attempts to introduce and further examine new modifications of Gamma and Kummer function in terms of Mittag-Leffler functions, respectively. Several attributes and formulations of this new Kummer-type function that include integral representations, Beta transform, Laplace transform, derivative formulas, and recurrence relation are investigated. Furthermore, outcomes of Riemann-Liouville fractional integral and fractional derivative in relation to this newly established Kummer function are also investigated.


Introduction
During the 20th century, theories of fractional differintegrals (fractional calculus (FC)) and the related special functions (SFs) have become marvellous tools in developing complex analysis. These theories also appear widely in a variety of disciplines of engineering, mathematics, and physics. FC is one of the outstanding disciplines of applied mathematics discussing the merits and implementations of noninteger (real or complex), order integrals and derivatives. It is a generalized version of classical (integerorder) calculus. This discipline involves left and right differ-integrals (correspondingly to left and right derivatives). The most frequently utilized fractional operators are right and left Caputo fractional derivative operators and right and left Riemann-Liouville integral and derivative operators. Actually, the differintegral is an operator that involves both integerorder derivatives and integrals as particular cases, which is why several implementations have become popular in the current FC. Particularly, it includes the principles and techniques of resolving differential equations with fractional derivatives of the unknown function, called fractional differential equations (FDEs). The history of FC was initiated at around the same time as when classical calculus was created by Newton and Leibniz in the 17th century. It was first stated in Leibniz's letter to L'Hospital in 1695, where the idea of semi-derivative was proposed. In other words, the idea of generalizing the derivative principle to non-integer order, especially to the order 1/ 2, is included in the correspondence of Leibniz and L'Hospital. FC was structured on original systematic bases by a multitude of mathematicians, for instance, Euler, Lagrange, Abel, Liouville, Riemann, Gr€ unwald, Laplace, Fourier, and others. For more details in chronological order, see (Oldham & Spanier, 1974) and (Podlubny, 1999).
In this context, SFs are specific mathematical functions that are crucial gadgets in advanced calculus and in almost all areas of mathematics. Several famed types of SFs are useful in solving various problems of FDEs. In fact, SFs have a considerable role in FC. As early as 1729, Euler provided the first fundamental function of FC, namely Gamma function, which is a generalized factorial formula that ranges from positive integers to complex values. This function is formulated by associating with a certain special function called the exponential function e z (Podlubny, 1999), as: t KÀ1 e t dt, ðK 2 C, <ðKÞ > 0Þ: (1) Afterwards, Legendre and Euler introduced another important function closely related to the Gamma function, the Beta function (Podlubny, 1999) as: (2) Further, On the other side, corresponding to CðKÞ, the Pochhammer symbol (rising factorial), denotes ðKÞ j , and is defined (Podlubny, 1999) by Since then, the interest in utilizing Gamma function in all of SFs has vastly increased till the present day. The remarkable generalizations of the exponential function in terms of Gamma function, the so-called Mittag-Leffler type functions (M-LTFs), performs an appealing role in the study of FC. More precisely, these functions are principally used to discuss solutions for FDEs by means of the Laplace transform technique. Such generalization motivates the upcoming research to provide more innovative ideas that yield various formulations of M-LTFs and fractional operators, (Bansal, Jolly, Jain, & Kumar, 2019;Choi, Parmar, & Chopra, 2020;Kumar, Singh, & Baleanu, 2018;Parmar, 2015;Rahman et al., 2019). Furthermore, derivations of physical phenomena of exponential nature could be determined by the physical laws via the M-LTF (power-law), (Bhatter, Mathur, Kumar, Nisar, & Singh, 2020;Djida, Mophou, &Area, 2019 andSaqib, Khan, &Shafie, 2019). Due to successful diverse applications for M-LTFs, correlated with FC, in physics and mathematic allied problems, several researchers prompted a lot of attention to the behaviour of M-LTFs and extended their outcomes to the complex domain, (for instance, see Al-Janaby, 2018; Al-Janaby & Ahmad, 2018; Al-Janaby & Darus, 2019 and Nisar, 2019).
The first appearance of Mittag-Leffler function (M-LF) of 1-parameter dates backs to G€ osta Magnus Mittag-Leffler (Mittag-Leffler, 1903) in 1903. Such series is proposed as: , ðz, b 2 C, <ðbÞ > 0Þ: (5) It is a generalization of e z . For b ¼ 1 in (5), it coincides with e z . Two years later, Wiman gave the following generalization of M-LF of 2-parameters as: which is the so-called Wiman function (or MLT-F) (Wiman, 1905a), and (Wiman, 1905b). The initial and lengthy studies focussed on the base merits of the MLT-Fs as entire functions and extended to the theoretical field of pure mathematics. After around three decades, the MLT-Fs implementation period was lastly achieved. In 1930, the authors Hille and Tamarkin, (1930) employed them in resolving the integral equations, namely Abel integral equations. Meanwhile, in 1947, Gross (1947 made use of the MLT-Fs to discuss the creep and relaxation functions. In 1954, by utilizing them, Barrett (1952) became a prominent pioneer in solving fractional differential equations. Then, in the year 1971, Caputo and Mainardi (1971) examined the fractional viscoelasticity using the MLT-Fs. During the continuous research in 1971, Prabhakar (1971) presented the MLT-F of 3-parameters, which is a more general formula of power series (6) and commonly used among FDEs with three or more terms, as: ðz, b, g, q 2 C, <ðbÞ > 0, <ðgÞ > 0, <ðqÞ > 0Þ: In 1995, Kilbas and Saigo (1995) proposed a generalized MLT-F to another 3-parameters, involving a special entire function as: ðz, b, 2 C, g, q 2 R, <ðbÞ > 0, <ðqÞ > 0Þ: Later, in 2009, Srivastava and Z. Tomovski (2009) introduced a more general MLT-F of 4-parameters as Other SFs, such as Wright and Kummer (confluent hypergeometric) functions, constructed by Gamma functions, are important in developing this regard. These functions are proposed, respectively, as: and This Wright function, correlating with the Mittag-Leffler function (6) was first formulated by Wright in 1933(Wright, 1933. The importance of the Wright function was demonstrated in solving a linear partial fractional differential equation, for instance, fractional diffusion-wave equation (Mainardi, 1996). Furthermore, there are several studies dedicated to employing this function in resolving the partial differential equation of the fractional-order extending the classical diffusion and wave equations (Luchko & Gorenflo, 1998). Whilst, the Kummer function was presented by Kummer (1837), it was presented as a solution to the second-order linear homogeneous differential equation: where #, ., z are unrestricted. This function has a fruitful role in diverse problems in physics.
Particularly, it is utilized as a solution to the differential equation for the velocity distribution function of electrons in a high-frequency gas discharge, see (MacDonald, 1949). In a recent time, Ghanim and Al-Janaby (2021a) imposed a new extension of generalized MLT-F and Kummer function of 4-parameters in another formula as: This function (13) is named Mittag-Leffler-Confluent hypergeometric functions (MLCHF). Note that E 1, g 1, g ðzÞ ¼ e z and E 1, 1 1, b ðzÞ ¼ E b ðzÞ were written by (5). Further E 1, .
On the other hand, in recent times, various generalizations and extensions of Gamma and Beta functions have been fruitfully put forward and presented. In 1994, Chaudhry and Zubair (1994) presented the extension of CðKÞ as follows: After that, in 1997, Chaudhry, Qadir, Rafique, and Zubair (1997) investigated the following extension of Beta functions ð<ðKÞ > 0, <ðwÞ > 0, <ðrÞ > 0Þ: They noted that C 0 ðKÞ ¼ CðKÞ and B 0 ðK, wÞ ¼ BðK, wÞ: In 2005, following a different methodology, Diaz and Teruel (2995) posed k-Gamma and k-Beta functions as a general formula for Gamma and Beta functions, respectively. They also studied a number of their merits. While, in 2007 and 2010, they discussed k-hypergeometric functions in terms of Pochhammer k-symbols for factorial functions (Diaz, Ortiz, & Pariguan, 2010), and (Diaz & Pariguan, 2007), respectively. Since then, these studies have attracted great interest by various investigators, Kokologiannaki (2010), Krasniqi (2010), Mansour (2009) and Merovci (2010) introduced and posed the scope of k-Gamma and k-Beta functions. Pursuing this type of study, in 2011, € Ozergin, € Ozarslan, and Altı n (2011) presented and examined a new extension of Gamma, Beta associated with hypergeometric functions as: They also studied new generalized Gauss hypergeometric function and confluent hypergeometric function. Moreover, they discussed some integral representations, differentiation properties, recurrence relations and summation formulas for these new generalized functions. In 2014, Srivastava, Çetinkaya, and Onur Kıymaz (2014) examined and introduced a certain generalized Pochhammer symbol along with its implementations based on hypergeometric function. These studies were followed by other complex analysts, who contributed to highlighting numerous new facets of this theme such as, Agarwal, Nieto, and Luo, (2017), € Ozarslan and Ustao glu (2019) and Rahman et al. (2020).
The implementation of fractional calculus in the physical model has succeeded in recent decades, the generalized M-LTFs were also used in mathematical and physical issues, as the solutions of the fractional integral and differential equations were naturally presented. Fractional order calculus is associated with practical endeavours, and it is widely used in nanotechnology (Baleanu, Guvenc, & Tenreiro Machado, 2010), chaos theory (Baleanu, Wu, & Zeng, 2017), optics (Esen, Sulaiman, Bulut, & Baskonus, 2018), human diseases  and other fields Taneco-Hern andez et al., 2019). In fact, the authors are collaborating with a group of college of engineering researchers on several recent engineering applications involving generalized multi-parameter Mittag-Leffler functions and their extended types, such as noise measurement and heat transfer in asphalt concrete.
Consequently, considering the aforementioned earlier works, this paper imposes new modifications of Gamma and Kummer functions based on Mittag-Leffler function, respectively. Several merits and formulations for this new generalized Kummer-type function which include integral representation, Beta transform, Laplace transform, derivative formulas, and recurrence relation are investigated. Moreover, outcomes associated with Riemann-Liouville fractional integral and fractional derivative for this considered generalization are also discussed.

Modified Gamma function
This section presents a new Gamma-type function based on MLT-F given by (13), namely the Gamma Mittag-Leffler function. It is an extension of the classical Gamma function written by (1). This function plays an important role in introducing the new Pochhammer symbol as well as the Kummer-type function in the next section.

Proposed Kummer-type function Q
l;˚;q;b g;};a;n ðzÞ This section imposes a fractional Pochhammer-Mittag-Leffler symbol, which is a general Pochhammer symbol (4). Moreover, it poses and discusses a significant special function called Kummer-type function in relation to this new Pochhammer symbol.
The following outcome presents the integral formulation for the function introduced in (26).
The next outcome provides a formal derivative for the new Kummer-type function coined in (26).

Conclusion
In light of the various employments of Fractional Calculus (FC) and Special Functions (SFs) in numerous areas of mathematics and applied science, here, a new extension of Gamma function by means of Mittag-Leffler type functions are introduced in a complex domain. Then, utilizing the extension of the Gamma function, a new general formula of the Pochhammer symbol is obtained. In addition, an interesting special function named Kummer-type function is presented. Certain related properties connected to this new Gamma, Pochhammer and Kummer functions involving integral representations, Beta transform, Laplace transform, derivative formulas, recurrence relation, and Riemann-Liouville fractional integral and fractional derivative are also investigated. For future research, the new functions and techniques achieved in this paper can be employed to enrich several areas of mathematics that include operator theory, inequalities theory, solving fractional differential equations along with developing a lot of realms of physics immensely.

Disclosure statement
No potential conflict of interest was reported by the author(s).