# Composition Formula for Saigo Fractional Integral Operator Associated with V-Function

Authors: Sunil Chandak; Anita Alaria; Biniyam Shimelis
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, we form integral formulas for Saigo’s hypergeometric integral operator involving V-function. Corresponding assertions for the classical Riemann–Liouville (R-L) and Erdélyi–Kober (E-K) fractional integral operator are extrapolated. Also, by putting in the transformations of Beta and Laplace, we can establish their composition formulas. By selecting the appropriate parameter values, the V-function may be reduced to a variety of functions, including the exponential function, Mittag–Leffler, Lommel, Struve, Wright’s generalized Bessel function, and Bessel and generalized hypergeometric function.


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## Body

## 1. Introduction and Preliminaries

Calculus with fractional orders is a branch of mathematics that develops from typical definitions of calculus with integer orders of integral and derivative operators, just how fractional exponents develop from integer exponents. In recent years, fractional calculus has been used in different fields of technology, research, plasma physics, economics, nonlinear control theory, applied maths, and bio-engineering. The V-function plays an important role to develop solutions to critical problem in terms of fractional-order integral and differential equations. The computations of fractional integrals and fractional derivatives involving transcendental functions of one and several variables are important because of the usefulness of their results, e.g., for evaluating differential and integral equations. The characteristics, execution, and various extensions of a number of fractional calculus operators have studied in depth by researchers (see [1–6]).We recollect the Saigo fractional integral operator containing Gauss’s hypergeometric functionF21. as a kernel. Let l,m,ξ∈ℂ and x>0; then, the generalized fractional integration operators related with Gauss hypergeometric function due to Saigo [7, 8] are defined as follows:(1)I0,xl,m,ξfx=x−l−mΓl∫0xx−zl−1F21l+m,−ξ;l;1−z/xfzdz,ℜl>0,(2)=dkdxkI0,xl+k,m−k,ξ−kfx,ℜl≤¯0,k=ℜ−l+1,(3)Ix,∞l,m,ξfx=1Γl∫x∞z−xl−1z−l−mF21l+m,−ξ;l;1−x/zfzdz,ℜl>0,(4)=−1kdkdxkIx,∞l−k,m−k,ξfx,ℜl≤¯0,k=ℜ−l+1.If we setm=0 in equations (1) and (3), we get the E-L fractional integral operators identified as(5)I0,xl,ξfx=x−l−ξΓl∫0xx−zl−1zξfzdz,ℜl>0,Ix,∞l,ξfx=xξΓl∫x∞z−xξ−1z−ξ−lfzdz,ℜl>0.The operatorsI0,xl,m,ξ and Ix,∞l,m,ξ include, as their special case, m=−l, the fractional integral operator of (R-L) and Weyl type as(6)I0,xl,−l,ξfx=I0,xlfx,Ix,∞l,−l,ξfx=Ix,∞lfx.For the present study, power function formulas of the fractional integral operators discussed above are needed as given in the following lemmas.Lemma 1.
Letx>0 and l,m,ξ,∈ℂ be such that ℜl>0; then, there exists the relation(7)I0,xl,m,ξzζ−1x=ΓζΓζ+ξ−mΓζ−mΓζ+ξ+lxζ−m−1ℜζ>max0,ℜm−ξ,(8)Ix,∞l,m,ξtζ−1x=Γ1−ζ+mΓ1−ζ+ξΓ1−ζΓ1−ζ+l+m+ξxζ−m−1,ℜζ<1+minℜm,ℜξ.Lemma 2.
Letx>0 and l,ξ,ζ∈ℂ be such that ℜl>0; then, there exists the relation(9)I0,xl,ξtζ−1x=Γζ+ξΓζ+ξ+lxζ−1,ℜζ>−ℜξ,Ix,∞l,ξtζ−1x=Γ1−ζ+ξΓ1−ζ+l+ξxζ−1,ℜζ<1+ℜξ.Lemma 3.
Letx>0 and l,ζ∈ℂ be such that ℜl>0; then, there exists the relation:(10)I0,xltζ−1x=ΓζΓζ+lxζ+l−1,ℜζ>0,Ix,∞ltζ−1x=Γ1−ζ+lΓ1−ζxζ+l−1,0<ℜζ<1−ℜl.Kumar [9] recently defined the V-function as follows:(11)Vz=Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,bu,ϕ,ν,σ;z=λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βz/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu,where(1)
L,κ,ω,y,ν,σ,kθθ=1,…,p,Avv=1,…,q,andguu=1,…,r are real numbers(2)
p,q, and r are natural numbers(3)
aθ,bv≥1θ=1,…,p;v=1,…,q(4)
ϕ>0,ℜβ>0,ℜd>0,z is a complex variable, and λ is an arbitrary constant(5)
The series on the RHS of (11) converges absolutely if q>p or q=p with Lz/2k≤1For descriptions of the series’ convergence constraints on (11) RHS, simply review [10–12]. The V-function defined by (11) is of a general character since it assimilates and applies a variety of valuable functions such as Macrobert’s E-function and exponential function [13], generalized Mittag–Leffler function [14–18], Lommel’s function, Struve’s function, generalized Bessel function and Bessel function [19–24], generalized hypergeometric function [13, 25, 26], and unified Riemann-zeta function [27].The purpose of this work is to evaluate the compositions of the generalized fractional integration operators (1) and (3) with the (11) V -function. Additionally, equivalent claims for the classical R-L and E-K fractional integral operators are evaluated. The conclusions mentioned in conjunction with the corollaries in theorems are undoubtedly innovative and conceptually valuable. Additionally, we derive the composition formulas for the findings stated in theorems using the Beta and Laplace transforms. Finally, as previously said, establish them as special instances on the primary result in conjunction with various special functions, the hypergeometric function, and so on.


## 2. Fractional Integration of V-Function

In this section, we develop image formulas for the V-function involving the Saigo fractional integral operator’s left and right sides. The following theorems give these formulas.Theorem 1.
Letl,m,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ>max0,ℜm−l. Then, the subsequent Saigo hypergeometric fractional integral I0,xl,m,ξ of V-function holds true:(12)I0,xl,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;zτx=λxζ−m−1∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βxτ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γζ+τnκ+dω+yΓζ+τnκ+dω+y+ξ−mΓζ+τnk+dω+q−mΓζ+τnκ+dω+y+ξ+l.Proof.
For convenience, using definitions (1) and (11), we denote the L. H. S. of the result (12) by ℑ1; by changing the order of summation and integration, we obtain(13)ℑ1=I0,xl,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;zτx=I0,xl,m,ξzζ−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βzτ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gux=λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−β∏v=1qbvn+Av∏u=1rdϕnσ+gu2nκ+dω+y×I0,xl,m,ξzζ+τnκ+dω+y−1x.
By applying relation (7) in (13), we get R.H.S. of (12).Theorem 2.
Letl,m,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ<1+minℜm,ℜξ. Then, the subsequent Saigo hypergeometric fractional integral Ix,∞l,m,ξ of V-function holds true:(14)Ix,∞l,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;z−τx=xζ−m−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βx−τ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γ1−ζ+τnκ+dω+y+mΓ1−ζ+τnκ+dω+y+ξΓ1−ζ+τnκ+dω+yΓ1−ζ+τnκ+dω+y+ξ+l+m.Proof.
To derive (14), we denote (14) by ℑ2 to L.H.S.; using definitions (3) and (11) and by changing the order of summation and integration, we obtain(15)ℑ2=Ix,∞l,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;z−τx=Ix,∞l,m,ξzζ−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βz−τ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gux=λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−β∏v=1qbvn+Av∏u=1rdϕnσ+gu2nκ+dω+y×Ix,∞l,m,ξzζ−τnκ+dω+y−1x.
By applying relation (8) in (15), we get the desired result (14).In relation, by settingm=−l and m=0, respectively, we work out the fractional integral formulas for the classical R-L and E-K fractional integral operators, which are stated by Corollaries 1 to 4.Corollary 1.
Letl,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ>0. Thus, the corresponding R-L fractional integral I0,xl of V-function holds true:(16)I0,xlzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;zτx=xζ+l−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βxτ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γζ+τnκ+dω+yΓζ+τnκ+dω+y+l.Corollary 2.
Letl,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ>−ℜξ. Therefore, the subsequent E-K fractional integral I0,xl,ξ of V-function holds true:(17)I0,xl,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;zτx=xζ−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βxτ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γζ+τnκ+dω+y+ξΓζ+τnκ+dω+y+l.Corollary 3.
Letl,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and 0<ℜζ<1−ℜl. Therefore, the subsequent R-L fractional integral Ix,∞l of V-function holds true:(18)Ix,∞lzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;z−τx=xζ+l−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βx−τ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γ1−ζ+τnκ+dω+y−lΓ1−ζ+τnκ+dω+y.Corollary 4.
Letl,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ>1+ℜξ. Therefore, the subsequent E-K fractional integral Ix,∞l of V-function holds true:(19)Ix,∞l,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;z−τx=xζ−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βx−τ/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γ1−ζ+τnκ+dω+y+ξΓ1−ζ+τnκ+dω+y+ξ+l.


## 3. Beta Transform of the Composition Formulas (12) and (14)

In this section, we develop some theorem that includes the results obtained in the previous section concerning the integral transformation.In this regard, we recall the definition of Beta function [28] as follows:(20)Bft;g,h=∫01tg−11−th−1ftdt,ℜg>0,ℜh>0.Theorem 3.
Letl,m,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1,ℜg>0, ℜh>0, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ>max0,ℜm−l. Therefore, the preceding fractional integral holds true:(21)BI0,xl,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;tzτx:g,h=xζ−m−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βxτt/2nκ+dω+y∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γζ+τ℘Γζ+τ℘+ξ−mΓb+℘ΓaΓζ+τ℘−mΓζ+τ℘+ξ+lΓb+℘+h,where ℘=nκ+dω+y.Proof.
On using (20), the L.H.S. of assertion (21) leads to(22)BI0,xl,m,ξzζ−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βA∏v=1qbvn+Av∏u=1rdϕnσ+gux;g,h,where(23)A=tzτ/2nκ+dω+y=∫01yg−11−yh−1×I0,xl,m,ξzζ−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βA∏v=1qbvn+Av∏u=1rdϕnσ+guxdy.
By interchanging the order of integration of the above term, it is reduced to(24)=λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−β∏v=1qbvn+Av∏u=1rdϕnσ+gu2℘×I0,xl,m,ξzζ+τ℘−1x∫01tg+τ℘−11−th−1dt.
By using the definition of Beta transform and (7), we obtain the R.H.S. of (21).Theorem 4.
Letl,m,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, ℜg>0, ℜh>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ<1+minℜm,ℜξ. Therefore, the preceding fractional integral holds true:(25)BIx,∞l,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;yz−τx=xζ−m−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βx−τ/2℘Γa∏v=1qbvn+Av∏u=1rdϕnσ+guΓb+℘+h×Γ1−ζ+τ℘+mΓ1−ζ+τ℘+ξΓb+℘Γ1−ζ+τ℘Γ1−ζ+τ℘+ξ+l+m.Proof.
The proof of Theorem4 is similar manner of Theorem 3.


## 4. Laplace Transform of the Composition Formulas (12) and (14)

In this section, we develop two theorems, including the Laplace transform, involving the result obtained in Section2 concerning the integral transform.The Laplace transform [28] of fz is defined as(26)Fs=Lfz;s=∫0∞e−stftdt,ℜs>0.Theorem 5.
Letl,m,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ℜg>0, ℜs>0, ϕ>0,ℜβ>0,ℜd>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ>max0,ℜm−l. Then, the subsequent fractional integral holds true:(27)Lyg−1I0,xl,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;yzτx=xζ−m−1sgλ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βx/sτ/2℘∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γζ+τ℘Γζ+τ℘+ξ−mΓb+τ℘Γζ+τ℘−mΓζ+τ℘+ξ+l.Proof.
On using (26), the left-hand side of assertion (27), leads to(28)Lyg−1I0,xl,m,ξzζ−1λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βyzτ/2℘∏v=1qbvn+Av∏u=1rdϕnσ+gux=∫0∞λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−β∏v=1qbvn+Av∏u=1rdϕnσ+gu2℘×I0,xl,m,ξzζ+τ℘−1xyg+τ℘−1e−sydy.
By interchanging the order of integration and summation, the right-hand side of (28) reduces to(29)=λ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−β∏v=1qbvn+Av∏u=1rdϕnσ+gu2℘×I0,xl,m,ξzζ+τ℘−1x∫0∞yg+τ℘−1e−sydy.
Calculating the integral and using (7), we get the R.H.S. of (27).Theorem 6.
Letl,m,ξ,ζ∈ℂ, L,κ,ω,y,ν,σ,kθ,Av,gu∈ℜ, p,q,r∈ℕ, aθ,bv≥1, ϕ>0,ℜβ>0,ℜd>0, ℜg>0, ℜs>0, and λ>0 is an arbitrary constant, such that ℜl>0 and ℜζ<1+minℜm,ℜξ. Then, the subsequent fractional integral holds true:(30)Lyg−1Ix,∞l,m,ξzζ−1Vnaθ,d,bvL,β,κ,ω,y,kθ,Av,gu,ϕ,ν,σ;yz−τx=xζ−m−1sgλ∑n=0∞−Ln∏θ=1paθn+kθd+ϕn+ν−βsx−τ/2℘∏v=1qbvn+Av∏u=1rdϕnσ+gu×Γ1−ζ+τ℘+mΓ1−ζ+τ℘+ξΓb+℘Γ1−ζ+τ℘Γ1−ζ+τ+℘+ξ+l+m.Proof.
The proof of Theorem6 is similar manner of Theorem 5.


## 5. Special Cases

In this section, we develop some new result as special cases of V-Function by selecting the appropriate parameter values in (12) and (14); we have some important results regarding the other special functions such as exponential function, Mittag–Leffler function, Lommel’s function, Struve’s function, Wright’s generalized Bessel function, Bessel function, and generalized hypergeometric function, given in the following corollaries:(1)
If we setd=1,u=1,L=−2,p=P,q=Q,κ=1,β=1,y=0,ω=0,kθ=0, Av=0,g1=−1,ϕ=1,ν=−1,σ=1, and λ=1 in (12) and (14), the V-function reduces to generalized hypergeometric function [13], using the result from equations (12) and (13), page 34, [29] as(31)ℑδn=δnδ∏j=1δℑ+j−1δn.under the condition stated in Theorems1 and 2, respectively, as(32)I0,xl,m,ξzζ−1FpQaP;bQ;zτx=xζ−m−1ΓζΓζ+ξ−mΓζ−mΓζ+ξ+l×P+2τFQ+2τa1,a2,…,aP,ζτ,ζ+1τ,…,ζ+τ−1τ,b1,b2,…,bQ,ζ−mτ,ζ−m−1τ,…,ζ−m+τ−1τ,ζ+ξ−mτ,ζ+ξ−m+1τ,…,ζ+ξ−m+τ−1τζ+ξ+lτ,ζ+ξ+l−1τ,…,ζ+ξ+l+τ−1τ|xτ,(33)Ix,∞l,m,ξzζ−1FpQaP;bQ;z−τx=xζ−m−1Γ1−ζ+mΓ1−ζ+ξΓ1−ζΓ1−ζ+ξ+l+m×P+2τFQ+2τa1,a2,…,aP,1−ζ+mτ,2−ζ+mτ,…,τ−ζ+mτ,b1,b2,…,bQ,1−ζτ,2−ζτ,…,τ−ζτ,1−ζ+ξτ,2−ζ+ξτ,…,ξ−ζ+ττ1−ζ+ξ+l+mτ,2−ζ+ξ+l+mτ,…,ξ−ζ+l+m+ττ|x−τ.(2)
If we takeu=1,τ=1,θ=1,v=2,a1=1,b1=1,b2=1,L=1,β=1,κ=2,ω=1,y=0,k1=0, a1=0, a2=0,b1=0,ϕ=1,ν=0,σ=1, and λ=1/Γd in (12) and (14), the V-function reduces to Bessel function [20], applying the result from equations (12) and (33), page 38, [29] as(34)θ2n=22nθ/2nθ+12n,under the condition stated in Theorems1 and 2, respectively, as below:(35)I0,xl,m,ξzζ−1Jdzx=Γζ+dΓζ+d+ξ−mxζ+d−m−1Γζ+d−mΓζ+d+ξ+lΓd+12d×F45ζ+d2,ζ+d+12,ζ+d+ξ−m2,ζ+d+ξ−m+12d+1,ζ+d−m2,ζ+d−m+12,ζ+d+ξ+l2,ζ+d+ξ+l+12|−x24,(36)Ix,∞l,m,ξzζ−1Jd1zx=Γm−ζ+d+1Γξ−ζ+d+1xζ−d−m−1Γ1−ζ+dΓl+m+ξ−ζ+d+1Γd+12d×F451−ζ+ξ+d2,ξ−ζ+d+22,1−ζ+d+m2,m−ζ+d+22d+1,1−ζ+d2,2−ζ+d2,l+m+ξ−ζ+d+12,l+m+ξ−ζ+d+22|−14x2.(3)
If we takeθ=1,v=2,u=1,a1=1,b1=1,b2=1,L=2,β=1,κ=1,ω=0,y=0,k1=0, A1=0, A2=0,g1=0,ν=0,σ=1, and λ=1/Γd in (12) and (14), the V-function reduces to Wright’s generalized Bessel function [20], using result (31) under the condition stated in Theorems 1 and 2, respectively:(37)I0,xl,m,ξzζ−1Jdϕzτx=xζ−m−1ΓζΓζ+ξ−mΓd+1Γζ−mΓζ+ξ+l×2τFϕ+2τζτ,ζ+1τ,…,ζ+τ−1τ,d+1ϕ,…,d+ϕϕ,ζ−mτ,ζ−m−1τ,…,ζ−m+τ−1τ,ζ+ξ−mτ,ζ+ξ−m+1τ,…,ζ+ξ−m+τ−1τζ+ξ+lτ,ζ+ξ+l−1τ,…,ζ+ξ+l+τ−1τ|xτϕϕ,(38)Ix,∞l,m,ξzζ−1Jdϕz−τx=xζ−m−1Γ1−ζ+mΓ1−ζ+ξΓd+1Γ1−ζΓ1−ζ+ξ+l+m×2τFϕ+2τ1−ζ+mτ,2−ζ+mτ,…,τ−ζ+mτ,d+1ϕ,…,d+ϕϕ,1−ζτ,2−ζτ,…,τ−ζτ,1−ζ+ξτ,2−ζ+ξτ,…,τ−ζ+ξτ1−ζ+ξ+l+mτ,2−ζ+ξ+l+mτ,…,τ−ζ+ξ+l+mτ|x−τϕϕ.(4)
If we takeu=1,τ=1,θ=1,v=2,a1=1,b1=3/2,b2=1,L=1,β=1,κ=2,ω=1,y=1,k1=0, A1=0,A2=0,g1=1/2,ϕ=1,ν=1/2,σ=1, and λ=1/ΓdΓ3/2 in (12) and (14), the V-function reduces to Struve’s function [20], using the result from (34) under the condition stated in Theorems 1 and 2, respectively, as below:(39)I0,xl,m,ξzζ−1Hdzx=Γζ+d+1Γζ+d+ξ−m+1xζ+d−mΓζ+d−m+1Γζ+d+ξ+l+1Γd+3/2Γ3/22d+1×F561,ζ+d+12,ζ+d+22,ζ+d+ξ−m+12,ζ+d+ξ−m+2232,d+32,ζ+d−m+12,ζ+d−m+22,ζ+d+ξ+l+12,ζ+d+ξ+l+22|−x24,(40)Ix,∞l,m,ξzζ−1Hd1/zx=Γm−ζ+d+2Γξ−ζ+d+2xζ−d−m−2Γ2−ζ+dΓl+m+ξ−ζ+d+2Γd+12d+1×F561,2−ζ+ξ+d2,ξ−ζ+d+32,2−ζ+d+m2,m−ζ+d+3232,d+32,2−ζ+d2,3−ζ+d2,l+m+ξ−ζ+d+22,l+m+ξ−ζ+d+32|−14x2.(5)
If we takeu=1,τ=1,θ=1,v=2,a1=1,k1=0,b1=δ′+υ′+3/2, b2=δ′−υ′+3/2,L=1,β=1,κ=2,d=1, ω=δ′,y=1,A1=0,A2=0,g1=−1,ϕ=1,ν=−1,σ=1, and λ=2δ′+1/δ′+υ′+1δ′−υ′+1 in (12) and (14), the V-function reduces to Lommel’s function [20], using the result from (34) under the condition stated in Theorems 1 and 2, respectively, as below:(41)I0,xl,m,ξzζ−1Sδ′,v′zx=Γζ+δ′+1Γζ+δ′+ξ−m+1xζ+δ′−mδ′+v′+1δ′−v′+1Γζ+δ′−m+1Γζ+δ′+ξ+l+1×F561,ζ+δ′+12,ζ+δ′+22,ζ+δ′+ξ−m+12,ζ+δ′+ξ−m+22δ′+v′+32,δ′−v′+32,ζ+δ′−m+12,ζ+δ′−m+22,ζ+δ′+ξ+l+12,ζ+δ′+ξ+l+22|−x24,(42)Ix,∞l,m,ξzζ−1Sδ′,v′1/zx=Γm−ζ+δ′+2Γξ−ζ+δ′+2xζ−δ′−m−2Γ2−ζ+δ′Γl+m+ξ−ζ+δ′+2×F561,2−ζ+ξ+δ′2,ξ−ζ+δ′+32,2−ζ+δ′+m2,m−ζ+δ′+32δ′+v′+32,δ′−v′+32,2−ζ+δ′2,3−ζ+δ′2,l+m+ξ−ζ+δ′+22,l+m+ξ−ζ+δ′+32|−14x2.(6)
If we takeu=1,τ=1,θ=1,v=1,a1=1,b1=1,L=−2,β=1,κ=1,ω=0,y=0,k1=0,a1=0,b1=−1, ν=−1,σ=1, and λ=1/Γd in (12) and (14), the V-function reduces to generalized Mittag–Leffler function [14, 16] using result (31), under the condition stated in Theorems 1 and 2, respectively, as below:(43)I0,xl,m,ξzζ−1Eϕ,dzx=ΓζΓζ+ξ−mxζ−m−1ΓdΓζ−mΓζ+ξ+l×3Fϕ+21,ζ,ζ+ξ−mdϕ,…,d+ϕ−1ϕ,ζ−m,ζ+ξ+l|xϕϕ,(44)Ix,∞l,m,ξzζ−1Eϕ,d1/zx=Γ1−ζ+mΓ1−ζ+ξxζ−m−1ΓdΓ1−ζΓ1−ζ+ξ+l+m×3Fϕ+21,1−ζ+m,1−ζ+ξdϕ,…,d+ϕ−1ϕ,1−ζ,1−ζ+ξ+l+m|1xϕϕ.(7)
If we takeu=1,τ=1,θ=1,v=1,d=1,a1=1,b1=1,L=2,κ=1,d=0,ω=0,y=0,k1=0,a1=0,b1=−1,ϕ=1, β=1,ν=−1,σ=1, and λ=1 in (12) and (14), the V-function reduces to exponential function under the condition stated in Theorems 1 and 2, respectively, as below:(45)I0,xl,m,ξzζ−1e−zx=ΓζΓζ+ξ−mxζ−m−1Γζ−mΓζ+ξ+l×F22ζ,ζ+ξ−mζ−m,ζ+ξ+l|−x,(46)Ix,∞l,m,ξzζ−1e−1/zx=Γ1−ζ+mΓ1−ζ+ξxζ−m−1Γ1−ζΓ1−ζ+ξ+l+m×F221−ζ+m,1−ζ+ξ1−ζ,1−ζ+ξ+l+m|−1x.


## 6. Concluding Remarks

In our present investigation, fractional integral operators and the V-function have advantage that it generalizes the R-L, Weyl, and Erdélyi–Kober fractional integral operators. Thus, we conclude the study by highlighting that many other important image formulas can be obtained as the special cases of our leading results (Theorems1 and 2) involving common fractional integral operators as stated above. Various special cases with similar outcomes of the report may be evaluated by taking acceptable values of the parameters concerned. We refer to [30, 31] for a variety of other special cases and give the results to interested readers. Our study ends with the remark that the stated outcome is important and can result in the yield of the number of other special function involving various types of Generalized hypergeometric function, Bessel function, Wright’s generalized Bessel function, Lommel’s function, Struve’s function, Mittag–Leffler, and exponential function.



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*Source: 2174708-2022-04-25.xml*