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Extended Unified Mittag-Leffler Function and Its Properties
Umar Muhammad ABUBAKAR1,*, Salim Rabi’u KABARA2, Ameer Abdullahi HASSAN3,
Faisal Adamu IDRIS4
1Kano University of Science and Technology, Wudil, Faculty of Computing and Mathematical
Sciences, Department of Mathematics, Kano State, Nigeria
uabubakar@kustwudil.edu.ng, ORCID:0000-0003-3935-4529
2KanoUniversity of Science and Technology, Wudil, Faculty of Computing and Mathematical
Sciences, Department of Mathematics, Kano State, Nigeria
srkabara@gmail.com, ORCID:0000-0002-7188-673X
3Kano University of Science and Technology, Wudil, Faculty of Computing and Mathematical
Sciences, Department of Mathematics, Kano State, Nigeria
ameernigeria@gmail.com, ORCID:0000-0002-8828-5332
4Sa’adatu Rimi College of Education, Kumbotso, Kano State, Nigeria
faisaladamidris@yahoo.com, ORCID: 0000-0003-3703-4936
Abstract
Special functions are an integral part of fractional calculus. In recent years, various extensions of
special functions such as gamma, beta, hypergeometric, Mittag-Leffler have been considered. In
this paper, an extended unified Mittag-Leffler function was introduced and some of its properties
were invistigated using a newly defined extended beta function. In addition, integral transforms
such as Laplace, Euler beta, Mellin, complex Fourier, Fourier sine and cosine, fractional Fourier,
Whittaker, Kemp-MacDonald, Fourier-Bessel and Varma transforms of the extended unified
Mittag-Leffler function are obtained. The convergence, recurrence relation and differential
formulas for this extended unified Mittag-Leffler function ware also discussed. However, Most
of the extended Q-functions and generalized Mittag-Leffler functions in the literature can be
deduced as special cases of this newly introduced extended unified Mittag-Leffler function.
Keywords: Q-function; Mittag-Leffler function, Integral transform, Recurence relation; Beta
function.
1. Introduction
The Andric eta al., [1] introduced the following generalization of Mittag-Leffler function:
(1)
where with
is pochhamer symbol and is the extended bata
function defined in [2-3].
Bhatnagar and Pandey [4] investigated the following generalized Q-function:
(2)
with
and is classical bata function (see, [5]).
Hang et al., [6] conbined the idea in (1) and (2) to proposed the following unified Mittag-Leffler
function:
(3)
Definition 1: For integrable function f on the Laplace transform is defined as [7]
where is the variable of transform.
Definition 2: The Euler beta transform is given by the following [8]:
for
Definition 3: The Mellin transform of integral function with index is defined by [9]
Definition 4: The complex Fourier transfrom is defined as follows [10]
Definition 5: The fractional Fourier transform is given [11-13]:
for
Definition 6: The cosine and sine Fourier transfroms are defined as [10]
and
Definition 7: The Hankel transform is expressed by the following [14, 15]
where and is Bessel-Maitland function (refer to, [7]).
Definition 8: For a complex parameter K-transform is defined by [14, 15]
and is MacDonald function or Modified Bessel function (see for example, [16]).
2. Main result
In this work, the following extended unified Mittag-Leffler function is introduced and
investigated:
(4)
if
and
is the extended beta function defined by [17]
for
3. Integral transforms
Throughout this section the following notation and assumption are adopted unless otherwise
stated:
Theorem 1: (Laplace transform)
Proof: By definition of Laplace transform, gives
Theorem 2: (Beta transform)
Proof: From the definition of beta transform,
Theorem 3: (Mellin transform)
Proof: Using definition of Mellin transform, yields
Using the relation in [18]
gives
Theorem 4: (Complex Fourier)
Proof: Applying the definition of complex Fourier transform, gives
On setting leads to
Theorem 5: (The Fourier cosine and sine transforms)
and
Proof By the definitions of Fourier cosine and sine transforms, yields
similarly,
Using the fact that
yields
On putting yields
hence
and
Theorem 6: (The fractional Fourier transform)
Proof: Applying the definition of fractional Fourier transform, gives
On setting
leads to
Theorem 7: (Whittaker transform)
where
and
Proof: Considering the improper integral
setting leads to
Applying the result [19]
, where
gives
where
and
Theorem 8: (Hankel transform)
where
Proof: Considering the integral
Applying [19]
gives
where
Theorem 9: (Kemp-MacDonald transform)
where
Proof: By applying the integral
Using the result [19]
leads to
where
Theorem 10: (Kemp-MacDonald)
where
Proof: Using the improper integral
Considering the equation [19]
gives
where
Theorem 11: (The Verma transform)
where
Proof:
setting leads to
Using the result [19]
, where
gives
where
Theorem 12: (The Verma transform)
where
Proof:
setting leads to
Using the result [19]
, where
gives
where
Theorem 13: (The Verma transform)
where
Proof:
setting leads to
Using the result [19]
, where
gives
where
4. Convergence of the unified extended Mittag-Leffler function
Theorem 14: the unified etended Mittag-Leffler function
converges
absolutely for all values of if
Proof: Using equation (4)
,
where
implies
and
applying the limit on both sides, yields
(5)
Considering the asymtotic behaviour for gamma function in [12]
One can obtain the following:
(6)
(7)
(8)
(9)
and
(10)
Applying equations (6)-(10), (5) can now be rewritten as
On simplifying, yields
The formula for the radius of convergence of a series is
.
Hence, the unified etended Mittag-Leffler function
converges
absolutely for all values of if
5. Recurrence relations and differential formulas
Theorem 15:
Proof:
Theorem 16:
Proof
Applying the result [20]
leads to
Using gives
Considering the properties of pochhammer symbol [21]
one can obtain
6. Conclusions
A new extended unified Mittag-Leffler function is proposed and some of its properties pertaing
integral transforms such as Euler beta, Laplace, Kemp-MacDonald, Hankel, Verma and
Whittaker transfroms are studied. This newly introduced extended unified Mittag-Leffler
function reduces to some well know generalized Mittag-Leffler and Q-function that exist in the
literature (see for example, Adric et al., [1], Bhatnagar and Pandey, [4], Hang et al., [6],
Sontakke et al., [18], Mazhar-Ul-Haque and Holambe, [20], Andric et al., [22] and Mazhar-Ul-
Haque and Holambe, [23]).
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