# Generalized Fractional Integral Formulas for thek-Bessel Function

Authors: D. L. Suthar; Mengesha Ayene
Journal: Journal of Mathematics
 (2018)

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## Abstract

The aim of this paper is to deal with two integral transforms involving the Appell function as their kernels. We prove some compositions formulas for generalized fractional integrals withk-Bessel function. The results are expressed in terms of generalized Wright type hypergeometric function and generalized hypergeometric series. Also, the authors presented some related assertion for Saigo, Riemann-Liouville type, and Erdélyi-Kober type fractional integral transforms.


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## Body

## 1. Introduction and Preliminaries

The generalization ofk-Bessel function is defined in Mondal [1] as(1)Wξ,ckz=∑n=0∞-cnΓknk+ξ+kn!z22n+ξ/k,where ξ>-1, k>0, and c∈R and Γk(z) is the k-gamma function defined in Dáaz and Pariguan [2] as(2)Γkz=∫0∞tz-1e-tk/kdt,z∈C.By inspection the following relation holds:(3)Γkz+k=zΓkz,and(4)Γkz=kz/k-1Γzk.If c=1 and k→1, then the generalized k-Bessel function defined in (1) reduces to the classical Bessel function Jv defined in Erdélyi [3]. For more details regarding k-Bessel function and its properties see [4–9].Here, we establish various generalized fractional integral formulas for thek-Bessel function. For this, we recall here the definition of generalized fractional integration operators of arbitrary order involving the Appell function F3 ([10], p.393, eq. (4.12) and (4.13)) as in the kernel. A generalization of the hypergeometric fractional integrals for υ,υ′,μ,μ′,η∈C and R(η)∈C is established by Marichev [11] as follows:(5)I0+υ,υ′,μ,μ′,ηfx=x-υΓη∫0xx-tη-1t-υ′F3υ,υ′,μ,μ′;η;1-tx,1-xtftdt,(6)I-υ,υ′,μ,μ′,ηfx=x-υ′Γη∫x∞t-xη-1t-υF3υ,υ′,μ,μ′;η;1-xt,1-txftdt.In (5) and (6), F3 denotes the Appell function (also well-known as Horn function) which is established by Srivastava and Karlsson [12].(7)F3υ,υ′,μ,μ′;η;x;y=∑p,q=0∞υpυ′qμpμ′qηp+qp!q!xpyqmax⁡x,y<1.The properties of this function are studied in Olver et al. ([13], P. (412-415)). Further relation for the Gauss hypergeometric functions exists as follows:(8)F3υ,ζ-υ,μ,ζ-μ;ζ;x;y=F21υ,μ,ζ;ζ;x+y-xy,The generalized hypergeometric type function pFq(z) is represent in Erdélyi [14] as(9)pFqz=pFqυ1,υ2,⋯,υpμ1,μ2,⋯,μq;z=∑n=0∞υ1nυ2n⋯υpnμ1nμ2n⋯μqnznn!.where υi,μj∈C; bj≠0,-1,-2,⋯, i=1,2,⋯,p, j=1,2,⋯,q, and (z)n is the Pochhammer symbols. Now, the gamma function is defined as(10)Γz=∫0∞tz-1e-tdt,λ∈C,(11)Γz+m=zmΓz,z∈C,and beta function is termed as(12)Bx,y=∫01tx-11-ty-1dt.The following series is defined in the Wright type hypergeometric function (see [15–17]) as(13)pΨqυi,Ai1,pμj,Bj1,q;z=pΨqz=∑n=0∞Γυ1+A1n⋯Γυp+ApnΓμ1+B1n⋯Γμq+Bqnznn!where μs and υr are positive real numbers, such that (14)1+∑s=1qμs-∑r=1pυr>0.Equation (13) differs from the generalized hypergeometric function pFq(z) defined in (9) only by a constant multiplier. The generalized hypergeometric function pFq(z) is a special case of pΨq(z) for Ai=Bj=1, where i=1,2,⋯,p and j=1,2,⋯,q:(15)1∏j=1qΓμjpFqυ1,⋯,υpμ1,⋯,μq;z=1∏i=1pΓυipΨqυi,11,pμj,11,q;z.For various properties of this function see [18].


## 2. Representation of Generalized Fractional Integrals in terms of Wright Functions

Marichev-Saigo-Maeda integrals operators were generalization of Saigo fractional integral operators [19]. In addition, their properties have been studied by Saigo and Maeda [10]. Considering this, the left-hand side and right-hand side of types (5) and (6) for a power function are as follows.Lemma 1.
(a) IfRη>0,Rδ>max0,Rυ+υ′+μ-η,Rυ′-μ′, then(16)I0+υ,υ′,μ,μ′,ηxδ-1x=ΓδΓδ+μ′-υ′Γδ+η-υ-υ,-μΓδ+μ′Γδ+η-υ′-μΓδ+η-υ-υ′T.(b) If Rη>0,Rδ<1+minR-μ,Rυ+υ′-η,Rυ+μ′-η, then(17)I-υ,υ′,μ,μ′,ηxδ-1x=Γ1-δ-η+υ+υ′Γ1-δ-μΓ1-δ+υ+μ′-ηΓ1-δΓ1-δ+υ+υ′+μ′-ηΓ1-δ+υ-μT,where T=xδ+η-υ-υ′-1.
The left-hand side generalized fractional integration (5) of the k-Bessel functions (1) is given by the following result.Theorem 2.
Lettingυ,υ′,μ,μ′,η,λ,ξ∈C be such that R(η)>0,R(ξ)>-1, R(λ+v+2nk)≥R(λ+ξ)>max⁡0,R(υ+υ′+μ-η),R(υ′-μ′), then the following formula holds: (18)I0+υ,υ′,μ,μ′,ηtλ/k-1Wξ,cktx=xλ+ξ/k+η-υ-υ′-12kλ/k×3Ψ4λ+ξk,2,λ+ξk-υ-υ,-μ+η,2,λ+ξk+μ′-υ′,2;λ+ξk+μ′,2,λ+ξk-υ-υ′+η,2,λ+ξk-υ′-μ+η,2,vk+1,1;-cx24k.Proof.
Using (1) and writing the function in the series form, the left-hand side of (18), leads to (19)I0+υ,υ′,μ,μ′,ηtλ/k-1Wξ,cktx=∑n=0∞-cn1/2ξ/k+2nΓkξ+k+nkn!×I0+υ,υ′,μ,μ′,ηtλ+ξ/k+2n-1x.Now, upon using the image formula (16), which is valid under the condition declared with Theorem 2, we get (20)I0+υ,υ′,μ,μ′,ηtλ/k-1Wξ,cktx=xλ+ξ/k-υ-υ′+η-12ξ/k∑n=0∞Γλ+ξ/k+2nΓλ+ξ/k+μ′+2n×Γλ+ξ/k-υ-υ,-μ+η+2nΓλ+ξ/k+μ′-υ′+2nΓλ+ξ/k-υ-υ′+η+2nΓλ+ξ/k-υ′-μ+η+2nΓξ/k+1+nkξ/k-cx2n4knn!.Using the definition of (15) in the right-hand side of (20), we arrive at the result (18).Special Cases of Theorem2(i) If we set υ′=0,μ=-γ,η=υ and replace υ by υ+μ in (18), then we get the following corollary relating to left-hand sided Saigo fractional integral operator ([19, 20]).Corollary 3.
Letυ,μ,γ,λ,ξ∈C, R(ξ)>-1, and R(λ+ξ+2nk)≥R(λ+ξ)>max⁡[0,R(μ-γ)]; then the following formula holds: (21)I0+υ,μ,γtλ/k-1Wξ,cktx=xξ/k+λ/k-μ-12kξ/k×2Ψ3λ+ξk,2,λ+ξk+γ-μ,2;λ+ξk-μ,2,λ+ξk+υ+γ,2,vk+1,1;-cx24k.(ii) If we set μ=-υ in (21), then we get the subsequent corollary relating to left-sided Riemann-Liouville type integral operator.Corollary 4.
Letυ,λ,ξ∈C be such that R(υ)>0, R(λ+ξ)>0, R(ξ)>-1; then the following result holds:(22)
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.(iii) If we set μ=0 in (21), then we get the subsequent corollary relating to left-sided Erdélyi-Kober type integral operator.Corollary 5.
Letυ,γ,λ,ξ∈C be such that R(υ)>0, R(λ+ξ+γ)>0,R(ξ)>-1; then the following formula holds: (23)Kυ,γ+tλ/k-1Wξ,cktx=xξ/k+λ/k-12kξ/k×1Ψ2λ+ξk+γ,2;λ+ξk+γ+υ,2,ξk+1,1;-cx24k.Theorem 6.
Lettingυ,υ′,μ,μ′,η,λ,ξ∈C be such that R(η)>0,R(ξ)>-1, R(λ-ξ-2n-1)≤1+R(λ-ξ-1)<1+min⁡R-μ,Rυ+υ′-η,Rυ+μ′-η, then the following formula holds: (24)I-υ,υ′,μ,μ′,ηtλ/k-1Wξ,ck1tx=xλ-ξ/k-υ-υ′+η-12kξ/k×3Ψ41+ξ-λk-μ,2,1+ξ-λk+υ+υ′-η,2,1+ξ-λk+υ+μ′-η,2;ξk+1,1,1+ξ-λk,2,1+ξ-λk+υ+υ′+μ′-η,2,1+ξ-λk+υ-μ,2;-c4kx2.Proof.
Using (2) and writing the function in the series form, the left-hand side of (24), leads to (25)I-υ,υ′,μ,μ′,ηtλ/k-1Wξ,ck1tx=∑n=0∞-cn1/2ξ/k+2nΓkξ+k+nkn!×I-υ,υ′,μ,μ′,ηtλ/k-ξ/k-2n-1x.Now, upon using the image formula (17), which is valid under the conditions declared with Theorem 6, we get (26)I-υ,υ′,μ,μ′,ηtλ/k-1Wξ,ck1tx=xλ/k-ξ/k-υ-υ′+η-12kν/k×∑n=0∞Γ1+ξ-λ/k+υ+υ′-η+2nΓ1+ξ-λ/k+υ+μ′-η+2nΓ1+ξ-λ/k+2nΓ1+ξ-λ/k+υ+υ′+μ′-η+2nΓ1+ξ-λ/k+υ-μ+2n×Γ1+ξ-λ/k-μ+2nΓξ/k+1+n-cn4kx2nn!.Using the definition of (15) in the right-hand side of (26), we arrive at the result (24).Special Cases of Theorem6(iv) If we substitute υ′=0,μ=-γ,η=υ and replace υ by υ+μ in (24), then we get the subsequent corollary relating to right-hand sided Saigo fractional integral operator [19].Corollary 7.
Lettingυ,μ,γ,λ,ξ∈C and R(ξ)>-1, R(υ)>0, R(λ-ξ)<1+min⁡[R(μ),R(γ)], then the following formula holds: (27)I-υ,μ,γtλ/k-1Wξ,ck1tx=xλ-ξ/k-μ-12kξ/k×2Ψ31+μ+ξ-λk,2,1+ξ-λk+γ,2;1+ξ-λk,2,1+μ+υ+γ+ξ-λk,2,ξk+1,1;-c4x2.(v) If we set μ=-υ in (27), then we get the following corollary relating to right-sided Weyl fractional type integral operator.Corollary 8.
Letυ,λ,ξ∈C be such that R(υ)>0, R(ξ)>-1, R(υ)<1-R(λ-ξ); then the following result holds: (28)I-υtλ/k-1Wξ,ck1tx=xξ/k-λ/k+υ-12kξ/k×1Ψ21-υ+ξ-λk,2;ξ-λk+1,2,ξk+1,1;-c4kx2.(vi) If we set μ=0 in (27), then we get the subsequent corollary relating to right-hand side of Erdélyi-Kober fractional type integral operator.Corollary 9.
Letυ,γλ,ξ∈C be such that R(ξ)>-1, R(υ)>0, R(λ+ξ)<1+min⁡[0,R(γ)]; then the following formula holds: (29)Kυ,γ-tλ/k-1Wξ,ck1tx=xξ/k-λ/k-12kξ/k×1Ψ21+ξ-λk+γ,2;1+ξ-λk+υ+γ,2,ξk+1,1;-c4kx2.


## 3. Representation in Terms of Generalized Hypergeometric Function

In this part, we introduce the generalized fractional integrals ofk-Bessel function in terms of generalized hypergeometric function. We consider the following well-known results:(30)Γ2z=22z-1πΓzΓz+12;z∈Cand(31)z2n=22nz2nz2+12n,z∈C,n∈NWe represent the following theorems containing the generalized hypergeometric function.Theorem 10.
Letυ,υ′,μ,μ′,η,λ,ξ∈C be such that R(η)>0, R(ξ)>-1, R(λ+ξ+2nk)≥R(λ+ξ)>max⁡0,R(υ+υ′+μ-η),R(υ′-μ′), and let λ+ξ/k,λ+ξ/k-υ-υ,-μ+η≠0,-1,⋯; then the following formula holds: (32)I0+υ,υ′,μ,μ′,ηtλ/k-1Wξ,cktx=xλ+ξ/k-υ-υ′+η-12kξ/k×Γλ+ξ/kΓλ+ξ/k-υ-υ,-μ+ηΓλ+ξ/k+μ′-υ′Γλ+ξ/k+μ′Γλ+ξ/k-υ′-υ+ηΓλ+ξ/k-υ′-μ+ηΓξ/k+1×F67λ+ξ2k,λ+ξ2k+12,λ+ξ2k+η-υ-υ,-μ2,λ+ξ2k+η-υ-υ,-μ+12,λ+ξ2k+μ′-υ′2,λ+ξ2k+μ′-υ′+12;ξk+1,λ+ξ2k+μ′2,λ+ξ2k+μ′+12,λ+ξ2k+η-υ-υ′2,λ+ξ2k+η-υ-υ′+12,λ+ξ2k+η-υ′-μ2,λ+ξ2k+η-υ′-μ+12;-cx24k.Proof.
Note thatF76 defined in (32) exit as the series is absolutely convergent. Now, using (11) with z=ξ/k+1 and (20) and applying (31) with z being replaced by λ+ξ/k, λ+ξ/k-υ-υ,-μ+η and λ+ξ/k+μ′-υ′, we have (33)I0+υ,υ′,μ,μ′,ηtλ/k-1Wξ,cktx=xλ+ξ/k-υ-υ′+η-12kξ/k×∑n=0∞Γλ+ξ/kΓλ+ξ/k-υ-υ,-μ+ηΓλ+ξ/k+μ′-υ′Γλ+ξ/k+μ′Γλ+ξ/k-υ-υ′+ηΓλ+ξ/k-υ′-μ+ηΓξ/k+1×λ+ξ/k2nλ+ξ/k-υ-υ,-μ+η2nλ+ξ/k+μ′-υ′2nλ+ξ/k+μ′2nλ+ξ/k-υ-υ′+η2nλ+ξ/k+η-υ′-μ2nξ/k+1n-cx2n4knn!,=xλ+ξ/k-υ-υ′+η-12kξ/kΓλ+ξ/kΓλ+ξ/k-υ-υ,-μ+ηΓλ+ξ/k+μ′-υ′Γλ+ξ/k+μ′Γλ+ξ/k-υ-υ′+ηΓλ+ξ/k-υ′-μ+ηΓξ/k+1×∑n=0∞λ+ξ/2knλ+ξ/2k+1/2nλ+ξ/2k+η-υ-υ,-μ/2nλ+ξ/2k+η-υ-υ,-μ+1/2nξ/k+1nλ+ξ/2k+μ′/2nλ+ξ/2k+μ′+1/2nλ+ξ/2k+η-υ-υ′/2nλ+ξ/2k+η-υ-υ′+1/2n×λ+ξ/2k+μ′-υ′/2nλ+ξ/2k+μ′-υ′+1/2nλ+ξ/2k+η-υ′-μ/2nλ+ξ/2k+η-υ′-μ+1/2n-cx2n4knn!.Thus, in accordance with (9), we get the required result (32).Corollary 11.
Letυ,μ,γ,λ,ξ∈C be such that R(υ)>0,R(ξ)>-1, R(λ+ξ)>max⁡[0,R(μ-γ)], and let λ+ξ/k,λ+ξ/k+γ-μ≠0,-1,⋯; then the following result holds:(34)
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.Corollary 12.
Letυ,λ,ξ∈C be such that R(υ)>0, R(λ+ξ)>0, R(ξ)>-1 and λ/k+ξ/k=0,-1,⋯; then the following result holds:(35)I0+υtλ/k-1Wξ,cktx=xλ/k+ξ/k+υ-12kξ/kΓλ+ξ/kΓ1+ξ/kΓλ+ξ/k-μ×2F3λ+ξ2k,λ+ξ2k+12;ξk+1,λ+ξ2k-μ2,λ+ξ2k-μ+12;-cx24k.Corollary 13.
Letυ,γ,ξ,λ∈C be such that R(υ)>0, R(λ+ξ)>0, R(ξ)>-1 and let λ/k+ξ/k+γ-μ≠0,-1,⋯; then the following result holds: (36)Kυ,γ+tλ/k-1Wξ,cktx=xξ/k+λ/k-12kξ/kΓλ+ξ/k+γΓ1+ξ/kΓλ+ξ/k+υ+γ×2F3λ+ξ2k+γ2,λ+ξ2k+γ+12ξk+1,λ+ξk+γ+υ2,λ+ξ2k+1+υ+γ2-cx24k.Theorem 14.
Lettingυ,υ′,μ,μ′,η,λ,v∈C be such that k>0, R(γ)>0,R(ν)>-1, R(λ-v-2n-1)≤1+R(λ-v-1)<1+min⁡R-μ,Rυ+υ′-η,Rυ+μ′-η, then there holds the following formula: (37)I-υ,υ′,μ,μ′,ηtλ/k-1Wv,ck1tx=xλ-ν/k+η-υ-υ′-12kν/k×Γ1+ν-λ/k-μΓ1+ν-λ/k-η+υ+υ′Γ1+ν-λ/k+υ+μ′-ηΓv/k+1Γ1+ν-λ/kΓ1+ν-λ/k+υ+υ′+μ′-ηΓ1+ν-λ/k+υ-μ×6F71+υ+υ′-η2+ν-λ2k,1+ν-λ2k+υ+υ′-η2,1+υ+μ′-η2+ν-λ2k,1+ν-λ2k+υ+μ′-η2,1-μ2+ν-λ2k,1+ν-λ2k-μ2;vk+1,1+υ+υ′+μ′-η2+ν-λ2k,1+ν-λ2k+υ+υ′+μ′-η2,12+ν-λ2k,1+ν-λ2k,1+υ-μ2+ν-λ2k,1+ν-λ2k+υ-μ2;-c4kx2.Proof.
Using (11) with z=v/k+1 and (26) and applying (31) with z being replaced by ν-λ/k-μ+1, ν-λ/k+υ+μ′-η+1, and ν-λ/k-η+υ+υ′+1, we have (38)I-υ,υ′,μ,μ′,ηtλ/k-1Wv,ck1tx=xλ-ν/k+η-υ-υ′-12kν/k×∑n=0∞Γ1+ν-λ/k-η+υ+υ′Γ1+ν-λ/k+υ+μ′-ηΓ1+ν-λ/k-μΓ1+ν-λ/kΓ1+ν-λ/k+υ+υ′+μ′-ηΓ1+ν-λ/k+υ-μΓv/k+1×1+ν-λ/k-η+υ+υ′2n1+ν-λ/k+υ+μ′-η2n1+ν-λ/k-μ2n1+ν-λ/k2n1+ν-λ/k+υ+υ′+μ′-η2n1+ν-λ/k+υ-μ2nv/k+1n-cn4kx2nn!,=xλ-ν/k+η-υ-υ′-12kν/kΓ1+ν-λ/k-η+υ+υ′Γ1+ν-λ/k+υ+μ′-ηΓ1+ν-λ/k-μΓ1+ν-λ/kΓ1+ν-λ/k+υ+υ′+μ′-ηΓ1+ν-λ/k+υ-μΓv/k+1×∑n=0∞1/2+ν-λ/2k+υ+υ′-η/2n1+ν-λ/2k+υ+υ′-η/2nv/k+1n1/2+ν-λ/2k+υ+υ′+μ′-η/2n1+ν-λ/2k+υ+υ′+μ′-η/2n×1/2+ν-λ/2k+υ+μ′-η/2n1+ν-λ/2k+υ+μ′-η/2n1/2+ν-λ/2k-μ/2n1+ν-λ/2k-μ/2n1/2+ν-λ/2kn1+ν-λ/2kn1+ν-λ/2k+υ-μ/2n1/2+ν-λ/2k+υ-μ/2n-cn4kx2nn!,Thus, in accordance with (9), we get the required result (37).Corollary 15.
Letυ,μ,γ,ξ,λ∈C and k>0 be such that R(υ)>0,R(ξ)>-1, R(v)>-1,R(υ)>0,R(λ-v)<1+min⁡[R(μ),R(γ)], and let μ-λ/k+ξ/k+1,γ-λ/k+ξ/k+1≠0,-1,⋯; then the following result holds: (39)I-υ,μ,γtλ/k-1Wξ,ck1tx=xλ/k-ξ/k-μ-12kξ/k×Γξ-λ/k+γ+1Γξ-λ/k+μ+1Γξ/k+1Γ1+ξ-λ/kΓξ-λ/k+υ+μ+η+1×4F5μ+12+ξ-λ2k,μ+22+ξ-λ2k,γ+12+ξ-λ2k,γ+22+ξ-λ2k;vk+1,12+ξ-λ2k,1+ξ-λ2k,υ+μ+γ+12+ξ-λk,υ+γ+22+ξ-λ2k;-c4kx2.Corollary 16.
Letυ,γ,ξ,λ∈C and k>0 be such that 0<R(υ)<1-R(λ-ξ), R(ξ)>-1, and let λ/k-ξ/k+υ≠1,2,⋯; then the following result holds:(40)I-υtλ/k-1Wξ,ck1tx=xλ/k-ξ/k+υ-12kξ/kΓ1-υ+ξ-λ/kΓξ/k+1Γ1+ξ-λ/k×2F31-μ2+ξ-λ2k,2-υ2+ξ-λ2k;ξk+1,12+ξ-λ2k,1+ξ-λ2k;-c4kx2.Corollary 17.
Letυ,γ,ξ,λ∈C and k>0 be such that R(ξ)>-1, R(υ)>0, R(λ+ξ)<1+max⁡[0,R(γ)] and let λ/k-ξ/k-γ≠1,2,⋯; then the following formula holds: (41)Kυ,γ-tλ/k-1Wξ,ck1tx=xλ/k-ξ/k-12kξ/kΓξ-λ/k+γ+1Γ1+ξ-λ/kΓξ/k+1×2F3γ+12+ξ-λ2k,γ+22+ξ-λ2k;ξk+1,υ+γ+12+ξ-λk,υ+γ+22+ξ-λ2k;-c4kx2.


## 4. Concluding Remark

MSM fractional integral operators have advantage that they generalize the R-L, Weyl, Erdélyi-Kober, and Saigo’s fractional integral operators; therefore, several authors called this a general operator. So, we conclude this paper by emphasizing that many other interesting image formulas can be derived as the specific cases of our leading results, Theorems2 and 6, involving familiar fractional integral operators as above said. Further, the generalized Bessel function defined in (1) possesses the lead that a number of Bessel functions, trigonometric functions, and hyperbolic functions happen to be the particular cases of this function. Some special cases of integrals involving generalized Bessel function have been explored in the literature by a number of authors ([20–26]) with different arguments. Therefore, results presented in this paper are easily converted in terms of a comparable type of novel interesting integrals with diverse arguments after various suitable parametric replacements.



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