# A New Class of Extended Hypergeometric Functions Related to Fractional Integration and Transforms

Authors: Vandana Palsaniya; Ekta Mittal; D. L. Suthar; Sunil Joshi
Journal: Journal of Mathematics
 (2022)

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## Abstract

The focus of this research is to use a new extended beta function and develop the extensions of Gauss hypergeometric functions and confluent hypergeometric function formulas that are presumed to be new. Four theorems have also been defined under the generalized fractional integral operators that provide an image formula for the extension of new Gauss hypergeometric functions and the extension of new confluent hypergeometric functions. Moreover, discussed are analogous statements in terms of the Weyl, Riemann–Liouville, Erdélyi–Kober, and Saigo fractional integral and derivative operator types. Here, we are also able to generate more image formulas by keeping some integral transforms on the obtained formulas.


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## Body

## 1. Introduction and Preliminaries

A special function is any of a number of mathematical functions that emerge in the solution of many classical physics problems. A recurrent theme in these challenges is the flow of electromagnetic, acoustic, and thermal energy. For the specific kind of special role of scientists and engineers, it has become an essential resource now a day. And in many fields, such as physical science, mathematical science, and engineering, this feature plays a very important role. There are numerous ways to define special functions. A series or an adequate integral can be used to define several specific functions of a complex variable. Special functions, such as Bessel functions, Whittaker functions, Gauss hypergeometric functions, and Jacobi, Legendre, Laguerre, and Hermite polynomials, have been continuously developed. Polynomial sequences are very important in applied mathematics (see [1–4]). Fractional calculus is a noninteger order generalisation of classical differentiation and integration. Riemann–Liouville, ErdéIyi–Kober, Hadamard, Caputo, Hilfer, Liouville–Caputo, Grünwald–Letnikov, Riesz, Coimbra, and Weyl all provided important definitions of fractional derivatives.Many substances’ consciousness and natural processes are characterised by fractional derivatives. As a generalisation of classical integer-order differential equations, fractional ordinary and partial differential equations exist. Fractional differential equations have been used to examine a variety of phenomena. It becomes more common to use it to illustrate problems in biology, electrodynamics, elasticity, viscoelasticity, fluid dynamics, physics, engineering, and a variety of other subjects (see [5–10]). Saigo [11] with generalized Gauss hypergeometric function (GHF) in the kernel and Saigo and Maeda [12] with Appell function in the kernel introduced generalized fractional integral operators.In our analysis, we have to remember the subsequent pair of Saigo–Maeda fractional integral operators.Letω,ω′,υ,υ′,ζ∈C such that Rˇζ>0, x>0, and the generalized fractional integral operator connecting Appell’s function or Horn’s function with the kernel [12] is defined as follows:(1)I0,xω,ω′,υ,υ′,ζfx=x−ωΓζ∫0xx−ℓζ−1ℓ−ω′F3ω,ω′,υ,υ′,ζ,1−ℓx,1−xℓfℓdℓ,(2)Jx,∞ω,ω′,υ,υ′,ζfx=xω′Γζ∫x∞ℓ−xζ−1ℓ−ωF3ω,ω′,υ,υ′,ζ,1−ℓx,1−xℓfℓdℓ.The Appell’s function or Horn’s functionF3⋅ in two variables [13] is defined as(3)F3ω,ω′,υ,υ′,ζ;x,y=∑m,s=0∞ωmω′sυmυ′sζm+sxmm!yss!,maxx,y<1.Remark 1.
The Appell’s functionF3⋅ diminishes into Gauss hypergeometric function F12 by the subsequent relation(4)F3ω,ζ−ω,υ,ζ−υ;ζ;x,y=F12ω,υ;ζ;x+y−xy,F3ω,0,υ,υ′;ζ;x,y=F12ω,υ;ζ;x,F30,ω′,υ,υ′;ζ;x,y=F12ω′,υ′;ζ;y.
Equations (1) and (2) reduce to Saigo operators, which are generalized fractional integral operators bound to the Gauss hypergeometric function if setting ω′=0,ω=ζ and ω′=0,ω=ω′=0 reduces it to the Saigo type, Erdélyi–Kober, Riemann–Liouville (R-L), and Weyl type fractional integral operator, respectively, as follows:(5)I0,xω,0,υ,υ′,ζfx=I0,xζ,ω−ζ,−υfx=x−ωΓζ∫0xx−ℓζ−1F12ω,υ;ζ;1−ℓxfℓdℓ,Jx,∞ω,0,υ,υ′,ζfx=Jx,∞ζ,ω−ζ,−υfx=1Γζ∫x∞ℓ−xζ−1ℓ−ωF12ω,υ;ζ;1−xℓfℓdℓ,where ζ>0,ω,υ∈R.
Moreover,(6)I0,xζ,0,υ,υ′,ζfx=E0,xζ,−υfx=x−ζ+υΓζ∫0xx−ℓζ−1ℓ−υfℓdℓ,Jx,∞ζ,0,υ,υ′,ζfx=Kx,∞ζ,−υfx=x−υΓζ∫x∞ℓ−xζ−1ℓ−ζ+υfℓdℓ,where ζ>0,υ∈R.
Moreover,(7)I0,x0,0,υ,υ′,ζfx=R0,xζfx=1Γζ∫0xx−ℓζ−1fℓdℓ,Jx,∞0,0,υ,υ′,ζfx=Wx,∞ζfx=1Γζ∫x∞ℓ−xζ−1fℓdℓ,where ζ>0.
The operatorI0,xω,ω′,υ,υ′,ζ− contains Saigo type, Riemann–Liouville (R-L), and Erdélyi–Kober fractional integral operator, and Jx,∞ω,ω′,υ,υ′,ζ− combines the Saigo type, Weyl, and Erdélyi–Kober fractional integral operator defined by Saigo [11].(i)
I0,xω,0,ζfx=E0,xω,ζfx, and I0,xω,−ω,ζfx=R0,xωfx(ii)
Jx,∞ω,0,ζfx=Kx,∞ω,ζfx and Jx,∞ω,−ω,ζfx=Wx,∞ωfxLemma 1.
Supposeω,ω′,υ,υ′,ζ,ℑ∈C and x>0 be s.t. Rˇζ>0, then the subsequent formula is valid(8)I0,xω,ω′,υ,υ′,ζℓℑ−1x=ΓℑΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υxℑ+ζ−ω−ω′−1,where Rˇℑ>max0,Rˇω′−υ′,Rˇω+ω′+υ−ζ.And(9)Jx,∞ω,ω′,υ,υ′,ζℓℑ−1x=Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υ×xℑ+ζ−ω−ω′−1,where Rˇℑ<1+minRˇ−υ,Rˇω+ω′−ζ,Rˇω+υ′−ζ.Lemma 2.
Supposeω,υ,ζ∈C be such that Rˇω>0; then, we have the subsequent relation(10)I0,xω,υ,ζℓℑ−1x=ΓℑΓℑ+ζ−υΓℑ−υΓℑ+ω+ζxℑ−υ−1,where Rˇℑ>max0,Rˇυ−ζ.And(11)Jx,∞ω,υ,ζℓℑ−1x=Γ1−ℑ+υΓ1−ℑ+ζΓ1−ℑΓ1−ℑ+ω+μ+ζxℑ−υ−1,where Rˇℑ<1+minRˇω,Rˇζ.Lemma 3.
Supposeω,ζ∈C be such that Rˇω>0; then, we have the subsequent relation(12)E0,xω,ζℓℑ−1x=Γℑ+ζΓℑ+ω+ζxℑ−1,Rˇℑ+ζ>0,Kx,∞ω,ζℓℑ−1x=Γ1−ℑ+ζΓ1−ℑ+ω+ζxℑ−1,Rˇζ<Rˇℑ>−1.Lemma 4.
Letω,ℑ∈C, then we have the subsequent relation(13)Rˇ0,xωℓℑ−1x=ΓℑΓℑ+ωxℑ+ω−1,Rˇℑ>0,Rˇω>0,Wx,∞ωℓℑ−1x=Γ1−ω−ℑΓ1−ℑxℑ+ω−1,Rˇℑ>Rˇω>−1.Supposehϑ=∑s=0∞aˇsϑs and sϑ=∑s=0∞bˇsϑs are the two series whose convergence radii are determined by Rˇh and Rˇs, respectively. Then, the series described by their Hadamard product [14] is(14)h∗sϑ=∑s=0∞aˇsbˇsϑs,where the radii of convergence R satisfy Rh,Rs⩽R.If one of the series specifies it and the radii of convergence of the other are greater than 0, the Hadamard series describes the complete function. Using the Hadamard product, we can deconstruct a developing function into two famous functions. For instance,Fq+raˇ,bˇ,υ,ωsϑ,σ can be crumbled as follows:(15)Fq+raˇ,bˇ,υ,ωsx1,…,xsy1,…,yq+r;ϑ;σ=Fr0−−y1,…,yr;ϑ;σ∗Fq+raˇ,bˇ,υ,ωsx1,…,xsy1+r,…,yq+r;ϑ;σ,ϑ<∞.


## 2. A Class of Extended Hypergeometric Functions

For the present investigation, we use the extension of the hypergeometric function specified by Palsaniya et al. [15] in the following way:(16)Fσ,δ;ϰMCaˇ,bˇ;cˇ;ϑ=∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇϑss!,(17)φσ,δ;ϰMCbˇ;cˇ;ϑ=∑s=0∞Bσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇϑss!,ϑ<1,Rˇaˇ>0,Rˇcˇ>Rˇbˇ>0,where Bσ,δ;ϰMCx,y extended beta function is defined as(18)Bσ,δ;ϰMCx,y=∫01ℓx−11−ℓy−1F11σ,δ;ϰℓ1−ℓdℓ,Rˇσ>0,Rˇδ>0,ϰ∈C,ϰ<K;0<K<2.0335.Now,Fσ,σ;ϰMCaˇ,bˇ;cˇ;ϑ=FϰMCaˇ,bˇ;cˇ;ϑ and Fσ,σ;0MCaˇ,bˇ;cˇ;ϑ=F12aˇ,bˇ;cˇ;ϑ(19)φσ,σ;ϰMCbˇ;cˇ;ϑ=φϰMCbˇ;cˇ;ϑ,φσ,σ;0MCbˇ;cˇ;ϑ=φ11bˇ;cˇ;ϑ.Ifϰ=0 and σ=δ, then equation (18) reduces to the classical beta function.Throughout this part, we use the latest extension of the generalized Gauss hypergeometric function (GHF)Fσ,δ;ϰMCaˇ,bˇ;cˇ;ϑ and the confluent hypergeometric function φσ,δ;ϰMCbˇ;cˇ;ϑ. If we take σ=δ in equations (16)–(18), the resultant definition was used by Chand et al. [16]. In the field of special functions, many kinds of extensions of Gamma and Beta functions are done by many researchers (see [17–24]).


## 3. Fractional Integration of the New Extended Hypergeometric Function

In this segment, we evaluate several fractional integral formulas for the generalized hypergeometric functionFσ,δ;ϰMCaˇ,bˇ;cˇ;ϑ and confluent hypergeometric function φσ,δ;ϰMCbˇ;cˇ;ϑ using the concept of the Hadamard product [14].Theorem 1.
Letω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C and x>0 be such that Rˇζ>0, ϰ∈C,ϰ<K;0<K<2.0335,then(20)I0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓx=xℑ+ζ−ω−ω′−1ΓℑΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+υ−ω−ω′Γℑ+ζ−ω′−υFσ,δ;ϰMCaˇ,bˇ;cˇ;x×F33ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′ℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ;x.Proof.
Let us consider that the left-hand side of equation (20) is symbolized by Iˇ. In equation (16), if we reverse the order of integration and summation, we get(21)Iˇ=I0,xω,ω′,υ,υ,ζℓℑ−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℓss!x,=∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!Iˇ0,xω,ω′,υ,υ′,ζℓs+ℑ−1x.
Using the result which is defined in (8), we have(22)=∑s=0∞aˇsΓs+ℑΓs+ℑ+ζ−ω−ω′−υΓs+ℑ+υ′−ω′Γs+ℑ+υ′Γs+ℑ+ζ−ω−ω′Γs+ℑ+ζ−ω′−υ×Bσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇxℑ+s+ζ−ω−ω′−1s!.
After simplification, using the property of the Pochhammer symbolσs=Γσ+s/Γσ, we obtain(23)=xℑ+ζ−ω−ω′−1ΓℑΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υ×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℑsℑ+ζ−ω−ω′−υsℑ+υ′−ω′sℑ+υ′sℑ+ζ−ω−ω′sℑ+ζ−ω′−υsxss!.
Finally, using the concept of the Hadamard product in the above equation, then it will change into the required result (20).Theorem 2.
Supposeω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 be such that Rˇζ>0, ϰ∈C,ϰ<K;0<K<2.0335, then(24)Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;1ℓx=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υ×Fσ,δ;ϰMCaˇ,bˇ;cˇ;1x×F331−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+υ′+ω1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ;1x.Proof.
Using the formula in equation (16), we designate the left side of equation (24) as Jˇ and then exchange the order of integration and summation, yielding the following result:(25)Jˇ=Jx,∞ω,ω′,υ,υ′,ζℓℑ−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1/ℓss!x=∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!Jˇx,∞ω,ω′,υ,υ′,ζℓℑ−s−1x.
Using the result of Lemma1 which is defined in equation (9), after some simplification, we obtain(26)=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υ×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1−ℑ−υs1−ℑ−ζ+ω+ω′s1−ℑ−ζ+υ′+ωs1−ℑs1−ℑ−ζ+ω+ω′+υ′s1−ℑ+ω−υs1/xss!.
Using the concept of Hadamard product, we get the desired result.Theorem 3.
Letω,ω′,υ,υ′,ζ,ℑ,σ,δ,bˇ,cˇ∈C and x>0 be such that Rˇω>0,Rˇcˇ>Rˇbˇ>0, ϰ∈C,ϰ<K;0<K<2.0335 such that Rˇℑ>max0,Rˇυ−ζ, then(27)I0,xω,ω′,υ,υ′,ζℓℑ−1φσ,δ;ϰMCbˇ;cˇ;ℓx=xℑ+ζ−ω−ω′−1ΓℑΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+υ−ω−ω′Γℑ+ζ−ω′−υφσ,δ;ϰMCbˇ;cˇ;x×F33ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′ℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ;x.Theorem 4.
Letω,ω′,υ,υ′,ζ,ℑ,σ,δ,bˇ,cˇ∈C and x>0 be such that Rˇω>0,Rˇcˇ>Rˇbˇ>0, ϰ∈C,ϰ<K;0<K<2.0335 such that Rˇℑ<1+minRˇυ,Rˇζ, then(28)Jx,∞ω,ω′,υ,υ′,ζℓℑ−1φσ,δ;ϰMCbˇ;cˇ;1ℓx=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υφσ,δ;ϰMCbˇ;cˇ;1x×F331−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+υ′+ω1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ;1x.Proof.
The proof of Theorems3 and 4 is as similar to that of Theorems 1 and 2, respectively.Remark 2.
If settingω′=0,ω=ζ and ω′=0,ω=ω′=0 in Theorems 1–4, then these will convert into the following interesting results asserted by the subsequent corollaries.Corollary 1.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C and x>0 be s.t. Rˇω>0,Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<k;0<K<2.0335, then(29)I0,xζ,ω−ζ,−υℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;ℓx=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓℑ+ζ−ωΓℑ+ζ−υ×Fσ,δ:ϰMCaˇ,bˇ;cˇ;x×F22ℑ,ℑ+ζ−ω−υℑ+ζ−ω,ℑ+ζ−υ;x.Corollary 2.
Letω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C and x>0 be such that Rˇω>0,Rˇℑ<1+minRˇυ,Rˇζ;minRˇσ,Rˇδ>0ϰ∈C,ϰ<K;0<K<2.0335, then(30)Jx,∞ζ,ω−ζ,−υℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;1ℓx=xℑ+ζ−ω−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓ1−ℑΓ1−ℑ+ω−υ×Fσ,δ;ϰMCaˇ,bˇ;cˇ;1x×F221−ℑ−υ,1−ℑ−ζ+ω1−ℑ,1−ℑ+ω−υ;1x.Corollary 3.
Letω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C and x>0 be such that Rˇω>0,Rˇcˇ>Rˇbˇ>0 and Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(31)I0,xζ,ω−ζ,−υℓℑ−1φσ,δ:ϰMCbˇ;cˇ;ℓx=xℑ+ζ−ω−1ΓℑΓℑ+ζ−ω−υΓℑ+ζ−ωΓℑ+ζ−υ×φσ,δ:ϰMCbˇ;cˇ;x×F22ℑ,ℑ+ζ−ω−υℑ+ζ−ω,ℑ+ζ−υ;x.Corollary 4.
Letω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C, x>0 be such that Rˇω>0,Rˇcˇ>Rˇbˇ>0 and Rˇℑ<1+minRˇυ⋅Rˇζ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(32)Jx,∞ζ,ω−ζ,−υℓℑ−1φσ,δ;ϰMCbˇ;cˇ;1ℓx=xℑ+ζ−ω−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓ1−ℑΓ1−ℑ+ω−υ×φσ,δ;ϰMCbˇ;cˇ;1x×F221−ℑ−υ,1−ℑ−ζ+ω1−ℑ,1−ℑ+ω−υ;1x.Remark 3.
Settingω=ζ in Corollaries 1–4 and employing the relation Saigo fractional integrals reduce to Erdélyi–Kober fractional integral operator.Remark 4.
If we replaceω=ω′=0 in Theorems 1–4, then Saigo fractional integral operators are reduced to Riemann–Liouville and Weyl fractional integral operators, respectively.Remark 5.
If we chooseϰ=0 and σ=δ in equations (16) and (17), the new extension of Gauss hypergeometric function Fσ,δ;ϰMCaˇ,bˇ;cˇ;ϑ and new extension of confluent hypergeometric function φσ,δ:ϰMCbˇ;cˇ;ϑ convert into classical function, and all results moreover reduce into fractional integration of classical functions in terms of Saigo–Maeda, Saigo, Erdélyi–Kober, Riemann–Liouville, etc.


## 4. Integral Transforms of Extended Gauss and Confluent Hypergeometric Functions

In this section, we demonstrate several theorems and corollaries by using different kinds of transforms like Beta transform and Laplace transform based on the results acquired in the previous section.


### 4.1. Beta Transform

The Beta transform offϑ is demarcated as(33)Bfϑ:aˇ,bˇ=∫01ϑaˇ−11−ϑbˇ−1fϑdϑ.Theorem 5.
Supposeω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C and x>0 be such that Rˇζ>0,Rˇℑ>max0,Rˇω+ω′+μ−ζ,Rˇω′−υ′;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(34)BI0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑx:l,m=xℑ+ζ−ω−ω′−1ΓmΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υΓl+mFσ,δ;ϰMCaˇ,bˇ;cˇ;ϑ×F44ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′,lℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ,l+m;x.Proof.
For convenience, we refer to the left-hand side of (34) by Bˇ; then applying the definition of beta transform, we have(35)Bˇ=∫01ϑl−11−ϑm−1I0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑxdϑ,and we may now use the formula in (16) and change the order of integration and summation to get the following result:(36)=∫01ϑl−11−ϑm−1I0,xω,ω′,υ,υ′,ζℓℑ−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℓϑss!xdϑ=∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!I0,xω,ω′,υ,υ′,ζℓs+ℑ−1∫01ϑl+s−11−ϑm−1dϑ.
By applying the formula given in (8) and using the definition of beta transform in the above equation, then it will reduce to(37)=∑s=0∞aˇs1s!Γs+ℑΓs+ℑ+ζ−ω−ω′−υΓs+ℑ+υ′−ω′Γs+ℑ+υ′Γs+ℑ+ζ−ω−ω′Γs+ℑ+ζ−ω′−υ×Bσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇxs+ℑ+ζ−ω−ω′−1Bl+s,m.
Further simplification of (37) yields(38)=xℑ+ζ−ω−ω′−1ΓmΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υΓl+m×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℑsℑ+ζ−ω−ω′−υsℑ+υ′−ω′slnℑ+υsℑ+ζ−ω−ω′sℑ+ζ−ω′−υsl+msxss!.
Interpreting (38) with the help of (16), we obtain(39)=xℑ+ζ−ω−ω′−1ΓmΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υΓl+m×F56aˇ,bˇ,ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′,lcˇ,ℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ,l+m;x.
Now, interpreting (39) with the view of the concept of the Hadamard product, we have the required result given in equation (34).Theorem 6.
Letω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 and Rˇζ>0, ϰ∈C,ϰ<K;0<K<2.0335, then(40)Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ϑℓx:l,m=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+ω+υ′ΓmΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υΓl+m×Fσ,δ:ϰMCaˇ,bˇ;cˇ;1xF441−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+ω+υ′,l1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ,l+m;1x.Proof.
We designate the left-hand side of (38) by Fˇ, for convenience, and then, using the concept of beta transform, we have(41)Fˇ=∫01ϑl−11−ϑm−1Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;ϑℓxdϑ.
Now, using the formula in (16) and reversing the integration and summation orders, we get(42)=∫01ϑl−11−ϑm−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇϑ/ℓss!Jx,∞ω,ω′,υ,υ′,ζℓℑ−1xdϑ=∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!Jˇx,∞ω,ω′,υ,υ′,ζℓℑ−s−1∫01ϑl+s−11−ϑm−1dϑ.
By applying the formula given in (9) and using the definition of beta transform in the above equation, then it will reduce to(43)=∑s=0∞Γ1−ℑ+s−υΓ1−ℑ+s−ζ+ω+ω′Γ1−ℑ+s−ζ+υ′+ωΓ1−ℑ+sΓ1−ℑ+s−ζ+ω+ω′+υ′Γ1−ℑ+s+ω−υ×aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!xℑ−s+ζ−ω−ω′−1Bl+s,m.
Further simplification of (43) yields(44)=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓmΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υΓl+m×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1−ℑ−υs1−ℑ−ζ+ω+ω′s1−ℑ−ζ+υ′+ωsls1−ℑs1−ℑ−ζ+ω+ω′+υ′s1−ℑ+ω−υsl+ms×1/xss!.
By interpreting (44), with the help of (16), we have(45)=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓmΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υΓl+m×F56aˇ,bˇ,1−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+υ′+ω,lcˇ,1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ,l+m;1x.
Now, interpreting (45) in the context of the Hadamard product concept, we arrive at the desired result, which is provided in (40).Corollary 5.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C be such that Rˇω>0,Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(46)BI0,xζ,ω−ζ,−υℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;ℓϑx:l,m=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓmΓℑ+ζ−ωΓℑ+ζ−υΓl+mFσ,δ:ϰMCaˇ,bˇ;cˇ;x×F33ℑ,ℑ+ζ−ω−υ,lℑ+ζ−ω,ℑ+ζ−υ,l+m;x.Corollary 6.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C be s.t. Rˇω>0,Rˇℑ<1+minRˇυ⋅Rˇυ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(47)BJx,∞ζ,ω−ζ,−υℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;1ℓx:,l,m=xℑ+ζ−ω−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓmΓ1−ℑΓ1−ℑ+ω−υΓl+mFσ,δ:ϰMCaˇ,bˇ;cˇ;1x×F331−ℑ−υ,1−ℑ−ζ+ω,l1−ℑ,1−ℑ+ω−υ,l+m;1x.Corollary 7.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C be s.t. Rˇω>0,Rˇcˇ>0,Rˇbˇ>0 and Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(48)BI0,xζ,ω−ζ,−υℓℑ−1φσ,δ:ϰMCbˇ;cˇ;ℓϑx:l,m=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓmΓℑ+ζ−ωΓℑ+ζ−υΓl+mφσ,δ:ϰMCbˇ;cˇ;x×F33ℑ,ℑ+ζ−ω−υ,lℑ+ζ−ω,ℑ+ζ−υ,l+m;x.Corollary 8.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C and x>0 and Rˇω>0,Rˇcˇ>Rˇbˇ>0 and Rˇℑ<1+minRˇυ⋅Rˇζ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(49)BJx,∞ζ,ω−ζ,−υℓℑ−1φσ,δ:ϰMCbˇ;cˇ;1ℓx:,l,m=xℑ+ζ−ω−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓmΓ1−ℑΓ1−ℑ+ω−υΓl+mφσ,δ:ϰMCbˇ;cˇ;1x×F331−ℑ−υ,1−ℑ−ζ+ω,l1−ℑ,1−ℑ+ω−υ,l+m;1x.


### 4.2. Laplace Transform

The Laplace transform offϑ is demarcated as(50)Lˇfϑ=∫0∞e−sϑfϑdϑ.Theorem 7.
Letω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C,x>0; Rˇω>0,Rˇℑ>max0,Rˇω+ω′+υ−ζ,Rˇω′−υ′;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(51)Lˇϑl−1I0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑx=xℑ+ζ−ω−ω′−1ΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υ1sl×F46aˇ,bˇ,ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′,lcˇ,ℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ;xs.Proof.
We designate the left-hand side of the result (51) by Lˇ, for convenience, and then using the definition of the Laplace transform, we obtain(52)Lˇ=∫0∞e−sϑϑl−1I0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑxdϑ.
Now using the formula given in (16) and then changing the order of integration and summation, we have(53)=∫0∞e−sϑϑl−1I0,xω,ω′,υ,υ′,ζℓℑ−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℓϑss!xdϑ.
By applying the formula given in (8) simplification of (53), then it will reduce to(54)=xℑ+ζ−ω−ω′−1∑s=0∞Γs+ℑΓs+ℑ+ζ−ω−ω′−υΓs+ℑ+υ′−ω′Γs+ℑ+υ′Γs+ℑ+ζ−ω−ω′Γs+ℑ+ζ−ω′−υ×aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇxss!∫0∞e−sϑϑl+s−1dϑ.
Putsϑ=u in equation (54) and applying the definition of the gamma function, we get(55)=xℑ+ζ−ω−ω′−1slΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υ×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇx/sss!ℑsℑ+ζ−ω−ω′−υsℑ+υ′−ω′slsℑ+υ′sℑ+ζ−ω−ω′sℑ+ζ−ω′−υs.
Using the concept of Hadamard product, we get the desired result.Theorem 8.
Supposeω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 be such that Rˇζ>0, ϰ∈C,ϰ<K;0<K<2.0335, then(56)Lˇϑl−1Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Bσ,δ:ϰMCaˇ,bˇ;cˇ;ϑℓx=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υ1sl×F46aˇ,bˇ,1−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+υ′+ω,lcˇ,1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ;1sx.Proof.
We designate the left-hand side of the result (56) by Lˇ, for convenience, then apply the Laplace transform, and obtain(57)Lˇ=∫0∞e−sϑϑl−1Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;ϑℓxdϑ.
Now, applying the formula given in (16) and then changing the order of integration and summation, we get(58)=∫0∞e−sϑϑl−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇϑss!Jx,∞ω,ω′,υ,υ′,ζℓℑ−s−1xdϑ.
By applying the formula given in equation (9), after simplification, it will be reduced to(59)=xℑ+ζ−ω−ω′−1∑s=0∞Γ1−ℑ+s−υΓ1−ℑ+s−ζ+ω+ω′Γ1−ℑ+s−ζ+υ′+ωΓ1−ℑ+sΓ1−ℑ+s−ζ+ω+ω′+υ′Γ1−ℑ+s+ω−υ×aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1/xss!∫0∞e−sϑϑl+s−1dϑ.
Putsϑ=u in (59) and using the definition of the gamma function, we have(60)=xℑ+ζ−ω−ω′−1slΓ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υ×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1−ℑ−υs1−ℑ−ζ+ω+ω′s1−ℑ−ζ+υ′+ωsls1−ℑs1−ℑ−ζ+ω+ω′+υ′s1−ℑ+ω−υs×1/sxss!.
Using the concept of Hadamard product, we get the desired result.Corollary 9.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 be s.t. Rˇω>0,Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(61)Lˇϑl−1I0,xζ,ω−ζ,−υℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑx=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓℑ+ζ−ωΓℑ+ζ−υ1sl×F35aˇ,bˇ,ℑ,ℑ+ζ−ω−υ,lcˇ,ℑ+ζ−ω,ℑ+ζ−υ;xs.Corollary 10.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 s.t Rˇω>0,Rˇℑ<1+minRˇυ⋅Rˇζ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(62)Lˇϑl−1Jx,∞ζ,ω−ξ,−υℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ϑℓx=xℑ−ω+ζ−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓ1−ℑΓ1−ℑ+ω−υ1sl×F35aˇ,bˇ,1−ℑ−υ,1−ℑ−ζ+ω,lcˇ,1−ℑ,1−ℑ+ω−υ;1sx.Corollary 11.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C,x>0, Rˇω>0,Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(63)Lˇϑl−1I0,xζ,ω−ζ,−υℓℑ−1φσ,δ;ϰMCbˇ;cˇ;ℓϑx=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓℑ+ζ−ωΓℑ+ζ−υ1sl×F34bˇ,ℑ,ℑ+ζ−ω−υ,lcˇ,ℑ+ζ−ω,ℑ+ζ−υ;xs.Corollary 12.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C and x>0 s.t. Rˇω>0,Rˇℑ<1+minRˇυ⋅Rˇζ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(64)Lˇϑl−1Jx,∞ζ,ω−ξ,−υℓℑ−1φσ,δ;ϰMCbˇ;cˇ;ϑℓx=xℑ−ω+ζ−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓ1−ℑΓ1−ℑ+ω−υ1sl×F34bˇ,1−ℑ−υ,1−ℑ−ζ+ω,lcˇ,1−ℑ,1−ℑ+ω−υ;1sx.


## 4.1. Beta Transform

The Beta transform offϑ is demarcated as(33)Bfϑ:aˇ,bˇ=∫01ϑaˇ−11−ϑbˇ−1fϑdϑ.Theorem 5.
Supposeω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C and x>0 be such that Rˇζ>0,Rˇℑ>max0,Rˇω+ω′+μ−ζ,Rˇω′−υ′;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(34)BI0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑx:l,m=xℑ+ζ−ω−ω′−1ΓmΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υΓl+mFσ,δ;ϰMCaˇ,bˇ;cˇ;ϑ×F44ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′,lℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ,l+m;x.Proof.
For convenience, we refer to the left-hand side of (34) by Bˇ; then applying the definition of beta transform, we have(35)Bˇ=∫01ϑl−11−ϑm−1I0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑxdϑ,and we may now use the formula in (16) and change the order of integration and summation to get the following result:(36)=∫01ϑl−11−ϑm−1I0,xω,ω′,υ,υ′,ζℓℑ−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℓϑss!xdϑ=∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!I0,xω,ω′,υ,υ′,ζℓs+ℑ−1∫01ϑl+s−11−ϑm−1dϑ.
By applying the formula given in (8) and using the definition of beta transform in the above equation, then it will reduce to(37)=∑s=0∞aˇs1s!Γs+ℑΓs+ℑ+ζ−ω−ω′−υΓs+ℑ+υ′−ω′Γs+ℑ+υ′Γs+ℑ+ζ−ω−ω′Γs+ℑ+ζ−ω′−υ×Bσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇxs+ℑ+ζ−ω−ω′−1Bl+s,m.
Further simplification of (37) yields(38)=xℑ+ζ−ω−ω′−1ΓmΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υΓl+m×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℑsℑ+ζ−ω−ω′−υsℑ+υ′−ω′slnℑ+υsℑ+ζ−ω−ω′sℑ+ζ−ω′−υsl+msxss!.
Interpreting (38) with the help of (16), we obtain(39)=xℑ+ζ−ω−ω′−1ΓmΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υΓl+m×F56aˇ,bˇ,ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′,lcˇ,ℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ,l+m;x.
Now, interpreting (39) with the view of the concept of the Hadamard product, we have the required result given in equation (34).Theorem 6.
Letω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 and Rˇζ>0, ϰ∈C,ϰ<K;0<K<2.0335, then(40)Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ϑℓx:l,m=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+ω+υ′ΓmΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υΓl+m×Fσ,δ:ϰMCaˇ,bˇ;cˇ;1xF441−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+ω+υ′,l1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ,l+m;1x.Proof.
We designate the left-hand side of (38) by Fˇ, for convenience, and then, using the concept of beta transform, we have(41)Fˇ=∫01ϑl−11−ϑm−1Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;ϑℓxdϑ.
Now, using the formula in (16) and reversing the integration and summation orders, we get(42)=∫01ϑl−11−ϑm−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇϑ/ℓss!Jx,∞ω,ω′,υ,υ′,ζℓℑ−1xdϑ=∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!Jˇx,∞ω,ω′,υ,υ′,ζℓℑ−s−1∫01ϑl+s−11−ϑm−1dϑ.
By applying the formula given in (9) and using the definition of beta transform in the above equation, then it will reduce to(43)=∑s=0∞Γ1−ℑ+s−υΓ1−ℑ+s−ζ+ω+ω′Γ1−ℑ+s−ζ+υ′+ωΓ1−ℑ+sΓ1−ℑ+s−ζ+ω+ω′+υ′Γ1−ℑ+s+ω−υ×aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1s!xℑ−s+ζ−ω−ω′−1Bl+s,m.
Further simplification of (43) yields(44)=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓmΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υΓl+m×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1−ℑ−υs1−ℑ−ζ+ω+ω′s1−ℑ−ζ+υ′+ωsls1−ℑs1−ℑ−ζ+ω+ω′+υ′s1−ℑ+ω−υsl+ms×1/xss!.
By interpreting (44), with the help of (16), we have(45)=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓmΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υΓl+m×F56aˇ,bˇ,1−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+υ′+ω,lcˇ,1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ,l+m;1x.
Now, interpreting (45) in the context of the Hadamard product concept, we arrive at the desired result, which is provided in (40).Corollary 5.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C be such that Rˇω>0,Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(46)BI0,xζ,ω−ζ,−υℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;ℓϑx:l,m=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓmΓℑ+ζ−ωΓℑ+ζ−υΓl+mFσ,δ:ϰMCaˇ,bˇ;cˇ;x×F33ℑ,ℑ+ζ−ω−υ,lℑ+ζ−ω,ℑ+ζ−υ,l+m;x.Corollary 6.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C be s.t. Rˇω>0,Rˇℑ<1+minRˇυ⋅Rˇυ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(47)BJx,∞ζ,ω−ζ,−υℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;1ℓx:,l,m=xℑ+ζ−ω−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓmΓ1−ℑΓ1−ℑ+ω−υΓl+mFσ,δ:ϰMCaˇ,bˇ;cˇ;1x×F331−ℑ−υ,1−ℑ−ζ+ω,l1−ℑ,1−ℑ+ω−υ,l+m;1x.Corollary 7.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C be s.t. Rˇω>0,Rˇcˇ>0,Rˇbˇ>0 and Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(48)BI0,xζ,ω−ζ,−υℓℑ−1φσ,δ:ϰMCbˇ;cˇ;ℓϑx:l,m=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓmΓℑ+ζ−ωΓℑ+ζ−υΓl+mφσ,δ:ϰMCbˇ;cˇ;x×F33ℑ,ℑ+ζ−ω−υ,lℑ+ζ−ω,ℑ+ζ−υ,l+m;x.Corollary 8.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C and x>0 and Rˇω>0,Rˇcˇ>Rˇbˇ>0 and Rˇℑ<1+minRˇυ⋅Rˇζ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(49)BJx,∞ζ,ω−ζ,−υℓℑ−1φσ,δ:ϰMCbˇ;cˇ;1ℓx:,l,m=xℑ+ζ−ω−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓmΓ1−ℑΓ1−ℑ+ω−υΓl+mφσ,δ:ϰMCbˇ;cˇ;1x×F331−ℑ−υ,1−ℑ−ζ+ω,l1−ℑ,1−ℑ+ω−υ,l+m;1x.


## 4.2. Laplace Transform

The Laplace transform offϑ is demarcated as(50)Lˇfϑ=∫0∞e−sϑfϑdϑ.Theorem 7.
Letω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C,x>0; Rˇω>0,Rˇℑ>max0,Rˇω+ω′+υ−ζ,Rˇω′−υ′;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(51)Lˇϑl−1I0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑx=xℑ+ζ−ω−ω′−1ΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υ1sl×F46aˇ,bˇ,ℑ,ℑ+ζ−ω−ω′−υ,ℑ+υ′−ω′,lcˇ,ℑ+υ′,ℑ+ζ−ω−ω′,ℑ+ζ−ω′−υ;xs.Proof.
We designate the left-hand side of the result (51) by Lˇ, for convenience, and then using the definition of the Laplace transform, we obtain(52)Lˇ=∫0∞e−sϑϑl−1I0,xω,ω′,υ,υ′,ζℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑxdϑ.
Now using the formula given in (16) and then changing the order of integration and summation, we have(53)=∫0∞e−sϑϑl−1I0,xω,ω′,υ,υ′,ζℓℑ−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇℓϑss!xdϑ.
By applying the formula given in (8) simplification of (53), then it will reduce to(54)=xℑ+ζ−ω−ω′−1∑s=0∞Γs+ℑΓs+ℑ+ζ−ω−ω′−υΓs+ℑ+υ′−ω′Γs+ℑ+υ′Γs+ℑ+ζ−ω−ω′Γs+ℑ+ζ−ω′−υ×aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇxss!∫0∞e−sϑϑl+s−1dϑ.
Putsϑ=u in equation (54) and applying the definition of the gamma function, we get(55)=xℑ+ζ−ω−ω′−1slΓℑΓlΓℑ+ζ−ω−ω′−υΓℑ+υ′−ω′Γℑ+υ′Γℑ+ζ−ω−ω′Γℑ+ζ−ω′−υ×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇx/sss!ℑsℑ+ζ−ω−ω′−υsℑ+υ′−ω′slsℑ+υ′sℑ+ζ−ω−ω′sℑ+ζ−ω′−υs.
Using the concept of Hadamard product, we get the desired result.Theorem 8.
Supposeω,ω′,υ,υ′,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 be such that Rˇζ>0, ϰ∈C,ϰ<K;0<K<2.0335, then(56)Lˇϑl−1Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Bσ,δ:ϰMCaˇ,bˇ;cˇ;ϑℓx=xℑ+ζ−ω−ω′−1Γ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υ1sl×F46aˇ,bˇ,1−ℑ−υ,1−ℑ−ζ+ω+ω′,1−ℑ−ζ+υ′+ω,lcˇ,1−ℑ,1−ℑ−ζ+ω+ω′+υ′,1−ℑ+ω−υ;1sx.Proof.
We designate the left-hand side of the result (56) by Lˇ, for convenience, then apply the Laplace transform, and obtain(57)Lˇ=∫0∞e−sϑϑl−1Jx,∞ω,ω′,υ,υ′,ζℓℑ−1Fσ,δ:ϰMCaˇ,bˇ;cˇ;ϑℓxdϑ.
Now, applying the formula given in (16) and then changing the order of integration and summation, we get(58)=∫0∞e−sϑϑl−1∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇϑss!Jx,∞ω,ω′,υ,υ′,ζℓℑ−s−1xdϑ.
By applying the formula given in equation (9), after simplification, it will be reduced to(59)=xℑ+ζ−ω−ω′−1∑s=0∞Γ1−ℑ+s−υΓ1−ℑ+s−ζ+ω+ω′Γ1−ℑ+s−ζ+υ′+ωΓ1−ℑ+sΓ1−ℑ+s−ζ+ω+ω′+υ′Γ1−ℑ+s+ω−υ×aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1/xss!∫0∞e−sϑϑl+s−1dϑ.
Putsϑ=u in (59) and using the definition of the gamma function, we have(60)=xℑ+ζ−ω−ω′−1slΓ1−ℑ−υΓ1−ℑ−ζ+ω+ω′Γ1−ℑ−ζ+υ′+ωΓlΓ1−ℑΓ1−ℑ−ζ+ω+ω′+υ′Γ1−ℑ+ω−υ×∑s=0∞aˇsBσ,δ;ϰMCbˇ+s,cˇ−bˇBˇbˇ,cˇ−bˇ1−ℑ−υs1−ℑ−ζ+ω+ω′s1−ℑ−ζ+υ′+ωsls1−ℑs1−ℑ−ζ+ω+ω′+υ′s1−ℑ+ω−υs×1/sxss!.
Using the concept of Hadamard product, we get the desired result.Corollary 9.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 be s.t. Rˇω>0,Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(61)Lˇϑl−1I0,xζ,ω−ζ,−υℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ℓϑx=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓℑ+ζ−ωΓℑ+ζ−υ1sl×F35aˇ,bˇ,ℑ,ℑ+ζ−ω−υ,lcˇ,ℑ+ζ−ω,ℑ+ζ−υ;xs.Corollary 10.
Supposeω,υ,ζ,ℑ,σ,δ,aˇ,bˇ,cˇ∈C, x>0 s.t Rˇω>0,Rˇℑ<1+minRˇυ⋅Rˇζ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(62)Lˇϑl−1Jx,∞ζ,ω−ξ,−υℓℑ−1Fσ,δ;ϰMCaˇ,bˇ;cˇ;ϑℓx=xℑ−ω+ζ−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓ1−ℑΓ1−ℑ+ω−υ1sl×F35aˇ,bˇ,1−ℑ−υ,1−ℑ−ζ+ω,lcˇ,1−ℑ,1−ℑ+ω−υ;1sx.Corollary 11.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C,x>0, Rˇω>0,Rˇℑ>max0,Rˇζ−υ;minRˇσ,Rˇδ>0;Rˇcˇ>0,Rˇbˇ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(63)Lˇϑl−1I0,xζ,ω−ζ,−υℓℑ−1φσ,δ;ϰMCbˇ;cˇ;ℓϑx=xℑ−ω+ζ−1ΓℑΓℑ+ζ−ω−υΓlΓℑ+ζ−ωΓℑ+ζ−υ1sl×F34bˇ,ℑ,ℑ+ζ−ω−υ,lcˇ,ℑ+ζ−ω,ℑ+ζ−υ;xs.Corollary 12.
Supposeω,υ,ζ,ℑ,σ,δ,bˇ,cˇ∈C and x>0 s.t. Rˇω>0,Rˇℑ<1+minRˇυ⋅Rˇζ;minRˇσ,Rˇδ>0 and ϰ∈C,ϰ<K;0<K<2.0335, then(64)Lˇϑl−1Jx,∞ζ,ω−ξ,−υℓℑ−1φσ,δ;ϰMCbˇ;cˇ;ϑℓx=xℑ−ω+ζ−1Γ1−ℑ−υΓ1−ℑ−ζ+ωΓlΓ1−ℑΓ1−ℑ+ω−υ1sl×F34bˇ,1−ℑ−υ,1−ℑ−ζ+ω,lcˇ,1−ℑ,1−ℑ+ω−υ;1sx.


## 5. Conclusion

In this present work, we developed new (presumed) extended Gauss hypergeometric functions (GHF) and confluent hypergeometric functions (CHF) which are presumed to be new. Few theorems have been defined which provide the image formulas for new extended Gauss hypergeometric functions (GHF) and confluent hypergeometric functions (CHF). Here, we are also able to generate more image formulas by keeping some integral transforms on the obtained formulas. With the help of the present work, we can further define the recurrence relations and integral representations which may be helpful in engineering and sciences.



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