# The Extended Bessel-Maitland Function and Integral Operators Associated with Fractional Calculus

Authors: Kelelaw Tilahun; Hagos Tadessee; D. L. Suthar
Journal: Journal of Mathematics
 (2020)

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## Abstract

The aim of this paper is to introduce a presumably and remarkably altered integral operator involving the extended generalized Bessel-Maitland function. Particular properties are considered for the extended generalized Bessel-Maitland function connected with fractional integral and differential operators. The integral operator connected with operators of the fractional calculus is also observed. We point out important links to known findings from some individual cases with our key outcomes.


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## Body

## 1. Introduction and Preliminaries

The Bessel-Maitland functionJϑς. is a generalization of Bessel function introduced by Ed. Maitland Wright [1] through a series representation as follows:(1)Jϑςz=∑m=0∞−zmΓςm+ϑ+1m!,z,ϑ∈ℂ,ς>0.In fact, Watson’s book [2] finds the application of the Bessel-Maitland function in the diverse field of engineering, chemical and biological sciences, and mathematical physics.Further, Pathak [3] defined generalization of the Bessel-Maitland function Jϑ,qς,δ. in the form as follows:(2)Jϑ,qς,δz=∑m=0∞δqm−zmΓςm+ϑ+1m!,where z∈ℂ\−∞,0;ς,ϑ,δ∈ℂ,ℜς≥0,ℜϑ≥−1,ℜδ≥0; q∈0,1∪ℕ, and δqm is known as generalized Pochhammer symbol which is defined as(3)δ0=1,δqm=Γδ+qmΓδ.Since the implementation of the Bessel-Maitland function in 1983, a number of extensions and generalizations have been introduced and examined with different applications (see information in [4–9]).By motivation of these investigations and applications of the Bessel-Maitland function, Suthar et al. [7] defined the generalized Bessel-Maitland function (2) in the following manner:(4)Jϑ,qς,δ;cz;p=∑m=0∞ℬpδ+qm,c−δcqm−zmBδ,c−δΓςm+ϑ+1m!,p>0,q∈ℕ,ℜc>ℜδ>0,which is known as extended generalized Bessel-Maitland function; here, ℬps,t is the extended beta function (see [10]).(5)ℬps,t=∫01zs−11−zt−1e−p/z1−zdz,ℜp>0,ℜs>0,ℜt>0.Forp=0, (5) reduces to beta function (see, e.g., [11], Section 1.1).Remark 1.
(i)
The particular case of equation (4), when p=0, reduces to (2) and when p=q=0, reduces to (1).(ii)
Whenq=1,ϑ=ϑ−1 and z=−z in (4) reduce to the extended Mittag–Leffler function defined by Ozarslan and Yilmaz ([12], equation (4)).Definition 1.
The space of Lebesgue measurable of real or complex valued functionLa,b for our study of the significance of fractional calculus is defined as follows:(6)La,b=f:f1=∫abfxdx<∞.Definition 2.
The Riemann–Liouville (R-L) fractional integral operatorsℑa+ℓ and ℑb−ℓ are defined respectively as (see, e.g., [13]) follows:(7)ℑa+ℓfx=1Γℓ∫axx−λℓ−1fλdλ,x>a,(8)ℑb−ℓfx=1Γℓ∫xbλ−xℓ−1fλdλ,x<b,where fx∈La,b,ℓ∈ℂ, and ℜℓ>0.Definition 3.
Forfx∈La,b;ℓ∈ℂ,ℜℓ>0 and n=ℜℓ+1, the Riemann–Liouville fractional differential operators Da+ℓ are defined by (see, e.g., [13])(9)Da+ℓfx=ddxnℑa+n−ℓfx.
Also,Da+ℓ,v of order 0<ℓ<1 and class 0<v<1 with reference to x which is the generalized form of (9) (see [13–15]) is defined as follows:(10)La+ℓ,vf=ℑa+v1−ℓddxℑa+1−v1−ℓfx.
On settingv=0 in (10), it reduces La+ℓ specified in (9) to the fractional differential operator.
We found the following baseline findings for our study.Lemma 1 (Mathai and Haubold [16]).
Ifℓ,u,∈ℂ,ℜℓ>0,ℜu>0, then(11)ℑa+ℓλ−au−1x=ΓuΓℓ+μx−aℓ+u−1.Lemma 2 (Srivastava and Manocha [17]).
If a functionfz is analytic and has a power series representation fz=∑m=0∞amzm in the disc z<ℜ, then(12)L0zℓzu−1fz=ΓuΓℓ+u∑m=0∞amumℓ+umzm.Lemma 3 (Srivastava and Tomovski [18]).
Letx>a,0<ℓ<1,0≤v≤1,ℜϑ>0 and ℜℓ>0. Then, the subsequent result holds true for La+ℓ,vf as follows:(13)La+ℓ,vλ−aϑ−1x=ΓϑΓϑ−ℓx−aϑ−ℓ−1.We also provided the subsequent established facts and rules in this article.Fubini’s theorem (Dirichlet formula) (Samko et al. [15])(14)∫abdz∫azfz,tdt=∫abdt∫tbfz,tdz.We define the following integral operator in terms of extended generalized Bessel function forδ,ω∈ℂ,ℜς>0 and ℜϑ>0 for our further analysis of fractional calculus, then the integral operator(15)ℑa+;ϑ,qω;ς,δ;cfx=∫axx−λϑJϑ,qς,δ;cωx−λς;pfλdλ,where x>a.If we putp=0 to the operator, then (15) reduces(16)ℑa+;ϑ,qω;ς,δfx=∫axx−λϑJϑ,qς,δωx−λςfλdλ.Ifω=0, then (16) reduces the integral operator to the R-L fractional integral operator described in (7).


## 2. Integral Operators with Extended Generalized Bessel-Maitland Function in the Kernel

In this part, we consider the composition of the fractional integral and derivative of Riemann–Liouville and the fractional derivative of Hilfer with the extended generalized Bessel-Maitland function defined by (4).Theorem 1.
Supposeς,ϑ,δ,c∈ℂ, ℜς>0,ℜϑ>−1,ℜc>ℜδ>0,p>0, and q,n∈ℕ, then the following result holds true:(17)ddznzϑJϑ,qς,δ;cωzς;p=zϑ−nJϑ−n,qς,δ;cωzς;p.Proof.
Using (4), we see that(18)ddznzϑJϑ,qς,δ;cωzς;p=ddznzϑ∑m=0∞ℬpδ+qm,c−δcqm−ωzςmBδ,c−δΓςm+ϑ+1m!=∑m=0∞ℬpδ+qm,c−δcqm−ωmBδ,c−δΓςm+ϑ+1m!ddznzςm+ϑ.
Using the identity,(19)ddxnxm=Γm+1Γm−n+1xm−n,m≥n,and after simplifying, we have(20)ddznzϑJϑ,qς,δ;cωzς;p=zϑ−n∑m=0∞ℬpδ+qm,c−δcqm−ωzςmBδ,c−δΓςm+ϑ−n+1m!.
Finally, it can be expressed by using (4) again, and we obtain(21)ddznzϑJϑ,qς,δ;cωzς;p=zϑ−nJϑ−n,qς,δ;cωzς;p.Corollary 1.
Supposeς,ϑ,δ∈ℂ, ℜς>0,ℜϑ>−1,ℜδ>0,p=0,q∈0,1∪ℕ, and n∈ℕ, then the following result holds true:(22)ddznzϑJϑ,qς,δωzς=zϑ−nJϑ−n,qς,δωzς.Theorem 2.
Ifx>aa∈ℜ+=0,∞,δ,ℓ,ϑ,ω∈ℂ,ℜϑ>−1,ℜℓ>0,p>0,q∈ℕ, then(23)ℑa+ℓλ−aϑJϑ,qς,δ;cωλ−aς;px=x−aϑ+ℓJϑ+ℓ,qς,δ;cωx−aς;p,(24)La+ℓλ−aϑJϑ,qς,δ;cωλ−aς;px=x−aϑ−ℓJϑ−ℓ,qς,δ;cωx−aς;p,(25)La+ℓ,vλ−aϑJϑ,qς,δ;cωλ−aς;px=x−aϑ−ℓJϑ−ℓ,qς,δ;cωx−aς;p,Proof.
(i)
Using (4) and (7), we have(26)ℑa+ℓλ−aϑJϑ,qς,δ;cωλ−aς;px=1Γℓ∫axλ−aϑJϑ,qς,δ;cωλ−aς;px−λℓ−1dλ=∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!1Γℓ∫axλ−aϑ+ςmx−λℓ−1dλ=∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!ℑa+ℓλ−aϑ+ςmx.By use of (11), we have(27)ℑa+ℓλ−aϑJϑ,qς,δ;cωλ−aς;px=∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!Γςm+ϑ+1Γςm+ϑ+ℓ+1x−aςm+ϑ+ℓ=x−aϑ+ℓ∑m=0∞ℬpδ+mq,c−δBδ,c−δcmqΓςm+ϑ+ℓ+1−ωx−aςmm!=x−aϑ+ℓJϑ+ℓ,qς,δ;cωx−aς;p.This occupies in the (23) statement.(ii)
On using (9), we have(28)La+ℓλ−aϑJϑ,qς,δ;cωλ−aς;px=ddxnℑa+n−ℓλ−aϑJϑ,qς,δ;cωλ−aς;px,and using (23), this takes the following form:(29)La+ℓλ−aϑJϑ,qς,δ;cωλ−aς;px=ddxnx−aϑ−ℓ+nJϑ−ℓ+n,qς,δ;cωx−aς;p.Applying (17), we get(30)La+ℓλ−aϑJϑ−ℓ,qς,δ;cωλ−aς;px=x−aϑ−ℓJς,ϑ−ℓγ,q;cωx−aς;p.This completes the desired proof (24).(iii)
By using (4), we obtain(31)La+ℓ,vλ−aϑJϑ,qς,δ;cωλ−aς;px=La+ℓ,v∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!λ−aςm+ϑx=∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!La+ℓ,vλ−aςm+ϑx.By applying (13), we get(32)La+ℓ,vλ−aϑJϑ,qς,δ;cωλ−aς;px=∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!Γςm+ϑ+1Γςm+ϑ−ℓ+1x−aςm+ϑ−ℓ=x−aϑ−ℓ∑m=0∞ℬpδ+mq,c−δBδ,c−δcmqΓςm+ϑ−ℓ+1−ωx−aςmm!=x−aϑ−ℓJϑ−ℓ,qς,δ;cωλ−aς;p,which brings in the necessary proof.Corollary 2.
Ifx>aa∈ℜ+=0,∞,δ,ℓ,ϑ,ω∈ℂ,ℜϑ>−1,ℜℓ>0,p=0,q∈0,1∪ℕ, then Theorem 2 reduces respectively to(33)ℑa+ℓλ−aϑJϑ,qς,δωλ−aςx=x−aϑ+ℓJϑ+ℓ,qς,δωx−aς,La+ℓλ−aϑJϑ,qς,δωλ−aςx=x−aϑ−ℓJϑ−ℓ,qς,δωx−aς,La+ℓ,vλ−aϑJϑ,qς,δωλ−aςx=x−aϑ−ℓJϑ−ℓ,qς,δωx−aς.


## 3. Some Properties of the Operatorℑa+;ϑ,qω;ς,δ;cfx

In this section, we derive several continuity properties of the generalized fractional integral operator.Theorem 3.
Ifδ,ω∈ℂ,ℜς>0,ℜϑ>−1,p>0,q∈ℕ, and ℜμ>0, then(34)ℑa+;ϑ,qω;ς,δ;cλ−aμ−1x=x−aμ+ϑΓμJϑ+μ,qς,δ;cωx−aς;p.Proof.
From (4) and (15), we obtain(35)ℑa+;ϑ,qω;ς,δ;cλ−aμ−1x=∫axx−λϑλ−aμ−1Jϑ,qς,δ;cωx−λς;pdλ=∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!∫axλ−aμ−1x−λϑ+ςmdλ=∑m=0∞Bpδ+mq,c−δcmq−ωmBδ,c−δm!1Γςm+ϑ+1×∫axλ−aμ−1x−λϑ+ςmdλ=∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δm!ℑa+ςm+ϑ+1λ−aμ−1x=∑m=0∞ℬpδ+mq,c−δcmqBδ,c−δ−ωmm!ΓμΓςm+ϑ+μ+1x−aςm+ϑ+μ=x−aϑ+μΓμJϑ+μ,qς,δ;cωx−aς;p.
This completes the desired proof.Corollary 3.
Ifδ,ω∈ℂ,ℜς>0,ℜϑ>−1,ℜℓ>0,q∈0,1∪ℕ, and ℜμ>0, then(36)ℑa+;ϑ,qω;ς,δλ−aμ−1x=x−aμ+ϑΓμJϑ+μ,qς,δωx−aς.Theorem 4.
Ifδ,ς,ϑ,ω,c∈ℂ,ℜς>0,ℜϑ>−1,ℜc>0,p>0, and q∈ℕ, then(37)ℑa+;ϑ,qω;ς,δ,cφ1≤Sφ1,where(38)S=b−aℜϑ+1∑m=0∞ℬpδ+mq,c−δcmqBδ,c−δΓςm+ϑ+1ℜϑ+ℜςm+1−ωb−aςmm!.Proof.
From (4), (6), and (15), we have(39)ℑa+;ϑ,qω;ς,δ;cφ1=∫abℑa+;ϑ,qω;ς,δ;cφxdx=∫ab∫axx−λϑJϑ,qς,δ;cωx−λς;pφλdλdx.
By exchanging the integration order and using Dirichlet formula (14), we have(40)ℑa+;ϑ,qω;ς,δ;cφ1≤∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!×∫ab∫λbx−λℜϑ+ℜςmdxφλdλ.
Settingu=x−λ, we obtain(41)ℑa+;ϑ,qω;ς,δ;cφ1≤∑m=0∞ℬpδ+mq,c−δcmq−ωmBδ,c−δΓςm+ϑ+1m!∫abuℜϑ+ℜςm+1ℜϑ+ℜςm+10b−a×φλdλ.
This can also be written as(42)ℑa+;ϑ,qω;ς,δ;cφ1≤b−aℜϑ+1∑m=0∞Bpδ+mq,c−δcmq−ωb−aςmBδ,c−δΓςm+ϑ+1ℜϑ+ℜςm+1m!.∫abφλdλ=Sφ1.
This completes the desired proof.Corollary 4.
Ifδ,ς,ϑ,ω,∈ℂ,ℜς>0,ℜϑ>−1, and q∈0,1∪ℕ, then(43)ℑa+;ϑ,qω;ς,δφ1≤ℌφ1,where(44)ℌ=b−aℜϑ+1∑m=0∞δmqΓςm+ϑ+1ℜϑ+ℜςm+1−ωb−aςmm!.Theorem 5.
Ifℓ,δ,ς,ϑ,ω∈ℂ,ℜς>0,ℜϑ>−1,ℜδ>0,ℜℓ>0,p>0,q∈ℕ, and x>a, for any function f∈Lς,ϑ, then the result holds true:(45)ℑa+ℓℑa+;ϑ,qω;ς,δ;cfx=ℑa+;ϑ+ℓ,qω;ς,δ;cfx=ℑa+;ϑ,qω;ς,δ;cℑa+ℓfx.Proof.
From (7) and (15), we have(46)ℑa+ℓℑa+;ϑ,qω;ς,δ;cfx=1Γℓ∫axx−λℓ−1ℑa+;ϑ,qω;ς,δ,cfλdλ=1Γℓ∫axx−λℓ−1∫aλλ−uϑJϑ,qς,δ;cωλ−uς;pfududλ.
By interchanging the order of integration and using (14), we have(47)ℑa+ℓℑa+;ϑ,qω;ς,δ;cfx=∫ax1Γℓ∫uxx−λℓ−1λ−uϑJϑ,qς,δ;cωλ−uς;pdλ×fudu.
Settingλ−u=ρ, we obtain(48)ℑa+ℓℑa+;ϑ,qω;ς,δ;cfx=∫ax1Γℓ∫0x−ux−u−ρℓ−1ρϑJϑ,qς,δ;cωρς;pdρfudu.
By applying (7) and (23), we get(49)ℑa+ℓℑa+;ϑ,qω;ς,δ;cfx=∫axx−uℓ+ϑJϑ+ℓ,qς,δ;cωx−uς;pfudu,thus, using (15), we get(50)ℑa+ℓℑa+;ϑ,qω;ς,δ;cfx=ℑa+;ϑ+ℓ,qω;ς,δ;cfx.
To demonstrate the second part, we start from the right side of (45), and using (7) and (15), we have(51)ℑa+;ϑ,qω;ς,δ;cℑa+ℓfx=∫axx−λϑJϑ,qς,δ;cωx−λς;pℑa+ℓfλdλ=∫axx−λϑJϑ,qς,δ;cωx−λς;p1Γℓ∫aλλ−uℓ−1fududλ=∫ax1Γℓ∫aλλ−uℓ−1x−λϑJϑ,qς,δ;cωx−λς;pfududλ.
By interchanging the order of integration and using (14), we obtain(52)ℑa+;ϑ,qω;ς,δ;cℑa+ℓfx=∫ax1Γℓ∫uxx−λϑλ−uℓ−1Jϑ,qς,δ;cωx−λς;pdλ×fudu.
Settingx−λ=ρ, we have(53)ℑa+;ϑ,qω;ς,δ;cℑa+ℓfx=∫ax1Γℓ∫0x−uρϑx−u−ρℓ−1Jϑ,qς,δ;cωρς;pdρfudu.
Again, by using (7) and applying (23), we get(54)ℑa+;ϑ,qω;ς,δ;cℑa+ℓfx=∫axx−uϑ+ℓJϑ+ℓ,qς,δ;cωx−uς;pfudu.
Finally, using (15), we obtain(55)ℑa+;ϑ,qω;ς,δ;cℑa+ℓfx=ℑa+;ϑ+ℓ,qω;ς,δ;cfx.
Thus, (50) and (55) complete the desired proof of (45).Corollary 5.
Ifℓ,δ,ς,ϑ,ω∈ℂ,ℜς>0,ℜϑ>−1,ℜδ>0,ℜℓ>0,q∈0,1∪ℕ, and x>a, for any function f∈Lς,ϑ, then Theorem 5 takes the form:(56)ℑa+ℓℑa+;ϑ,qω;ς,δfx=ℑa+;ϑ+ℓ,qω;ς,δfx=ℑa+;ϑ,qω;ς,δℑa+ℓfx.


## 4. Concluding Remark and Discussion

The newly defined integral operators involving the extended generalized Bessel-Maitland function is investigated here. Various special cases of the paper’s related results may be analyzed by taking appropriate values of the relevant parameters. For example, as given in remarks (i) and (ii), we obtain the undeniable result due to Gauhar et al. [19, 20]. For a number of other special cases, we refer to [21] and leave the findings to interested readers.



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