Integral  Transforms  through  Mellin-Barnes  Integration

Oleg Marichev & Paco Jain, Wolfram Technology Conference, November 3, 2023

Practice

motivation

Integral transforms are used extensively in the physical sciences and engineering, with applications to differential equations, integral equations, signal processing, etc. The plot below counts the appearance of different transforms in a subset of recent publications.

data=;
ListLogLogPlot[Callout[Length[#2],#1]&@@@Normal[Take[data, 100]],
                                AspectRatio->0.6, AxesLabel->{"Log[rank]","Log[num papers]"},ImageSize -> 600,ImagePadding->50]

The calculational challenge is to evaluate integrals of the form

(∫_a^b)f[τ] K[τ, t; {(Subscript[p, 1]), (Subscript[p, 2]), …,(Subscript[p, k])}]\[DifferentialD]τ

example application (electric potential inside an infinite wedge: Mellin transform)

Pictured above is an infinite wedge. The electrostatic potentials at θ=α and θ=-α are fixed functions of r, f[r] and g[r], respectively. In terms of polar coordinates in the x-y plane, the potential obeys the Laplace equation

(r^2) (Subscript[ϕ, rr]) + r (Subscript[ϕ, r]) + (Subscript[ϕ, θθ]) ==0

Applying the Mellin transform ℳ, defined by

ℳ[ϕ[r, θ]]== (OverBar[ϕ])[p, θ] == (∫_0^∞)(r^(p-1))ϕ[r, θ] \[DifferentialD]r

to this equation yields

(Subscript[(OverBar[ϕ]), θθ])+(p^2)(OverBar[ϕ]) == 0

with

(OverBar[ϕ])[p, α] == (OverBar[f])[p]

(OverBar[ϕ])[p, -α] == (OverBar[g])[p]

A general solution to this differential equation is

(OverBar[ϕ])[p, θ]== A Cos[p θ] + B Cos[p θ]

with A and B determined from the boundary conditions according to

A Cos[p α] + B Sin[p α]==(OverBar[f])[p]

A Cos[p α] - B Sin[p α]==(OverBar[g])[p]

Consider the case f[r]==g[r]==Exp[-r] Sin[r]. Then,

(OverBar[f])[p]==(OverBar[g])[p]==ResourceFunction["GenericIntegralTransform"][Exp[-r]Sin[r],r, p, "Mellin"]

%//Activate

which implies that

A== (((OverBar[f])[p]+(OverBar[g])[p])/(2 Cos[p α])), B == 0

or,

(OverBar[ϕ])[p, θ]==((((2^(p/2)) Gamma[(3/2)+(1/2) (-2+p)] Gamma[(3/2)+(1/2) (-1+p)] Sin[((p π)/4)])/(p (Sqrt[π])))/(Cos[p α]))Cos[ p θ]

The potential in terms of r and θ is then obtained by applying the inverse Mellin transform to this result (requires future FoxH technology).

main result

The result of this work is a Wolfram Language resource function for computing a variety of integrals of this form:

[]

Two special functions will be be of principle importance: MeijerG and FoxH. Transforms are calculated by representing integrands in terms of these functions, and results are returned in terms of them. Currently the MeijerG technology is well-fleshed out, but FoxH technology requires development.

table of transforms

transformList={"FourierCos","FourierSin","FourierExp",{"Hankel",\[FormalNu]},{"Hankel",\[FormalNu],\[FormalAlpha]},"Hartley","Hilbert",{"Hilbert",\[FormalAlpha]},{"Integrate",\[FormalAlpha]},"KontorovichLebedev","Laplace",{"Laplace",\[FormalAlpha]},"LaplaceTwoSided","Mellin",{"Mellin",\[FormalAlpha]},{"Neumann",\[FormalNu]},{"Neumann",\[FormalNu],\[FormalAlpha]},{"Riesz",\[FormalAlpha]},{"Stieltjes",\[FormalRho]},{"Struve",\[FormalNu]},{"Struve",\[FormalNu],\[FormalAlpha]},{"Weyl",\[FormalAlpha]}};
Length[transformList]

transGrid=Grid[
Prepend[{#,[1/(1+z),z, t, #],[Exp[-z],z, t, #]}&/@transformList,{"transform",HoldForm[ [1/(1+z),z, t,"transform"]],HoldForm[[Exp[-z],z, t, "transform"]]}]
,
Frame->All
]

transGrid//Activate//FunctionExpand

Theory

Introduction

(a^b)   ->  (x^b)  (*power*)   &  (a^(x   ))(*exponent*)  ( (k^b), (a^k) )

MellinTransform[f[x],x,s]==(∫_0^∞)(x^(s-1))f[x]\[DifferentialD]x==F[s]

LaplaceTransform[f[x],x,p]==(∫_0^∞)(E^(-p x))f[x]\[DifferentialD]x==F[p]

FourierTransform[f[x],x,p]==(1/(Sqrt[2π]))(∫_(-∞)^∞)(E^(I ω x))f[x]\[DifferentialD]x==F[ω]

with  corresponding  inversion  formulas

InverseMellinTransform[F[s],s,x]==(1/(2π I))(∫_(γ-i∞)^(γ+i∞))F[s](x^(-s))\[DifferentialD]s==f[x]

InverseLaplaceTransform[f[x],x,p]==(1/(2π I))(∫_(γ-i∞)^(γ+i∞))F[p](E^(p x))\[DifferentialD]p==f[x]

InverseFourierTransform[F[ω],ω,x]==(1/(Sqrt[2π]))(∫_(-∞)^∞)F[ω](E^(-I ω x))\[DifferentialD]x==f[x]

Mellin  transform  is  equivalent  to  Fourier  transform:

FourierTransform[f[t],t,x]==(1/(Sqrt[2 π]))MellinTransform[f[-Log[t]],t,-I x]

MellinTransform[f[t],t,x]==(Sqrt[2 π])FourierTransform[f[(E^(-t))],t,I x]

The Fourier  transform can be presented as combination of Cosine and Sine Fourier transforms:

FourierTransform[f[t],t,z]==FourierCosTransform[((f[t]+f[-t])/2),t,z]+I FourierSinTransform[((f[t]-f[-t])/2),t,z]

FourierCosTransform[f[t],t,z]==(Sqrt[(2/π)])(∫_0^∞)f[t]Cos[t z]\[DifferentialD]t (*==InverseFourierCosTransform[f[t],t,z]*)

FourierSinTransform[f[t],t,z]==(Sqrt[(2/π)])(∫_0^∞)f[t]Sin[t z]\[DifferentialD]t (*==InverseFourierSinTransform[f[t],t,z]*)

The Laplace  transform can be presented as Fourier transforms of the truncated function:

LaplaceTransform[f[t],t,z]==(Sqrt[2 π])FourierTransform[UnitStep[t]f[t],t,I z]

Example: Laplace transform

In terms of ResourceFunction by definition we have the following representations

[]

[(1/(1+x)),x,y,"Laplace"]//Activate//FunctionExpand

{[(1/(1+x)),x,Catalan,"Laplace"],LaplaceTransform[(1/(1+x)),x,Catalan]}

%//Activate//FunctionExpand

Mellin-Barnes integrals defining FoxH & MeijerG functions

Below contour integral by special contour ℒ  is called Mellin-Barnes integral. It defines FoxH function (1961)

FoxH[{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}]},{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},z]==
FoxH[{{{(Subscript[a, 1]),(Subscript[α, 1])},…,{(Subscript[a, n]),(Subscript[α, n])}},{{(Subscript[a, n+1]),(Subscript[α, n+1])},…,{(Subscript[a, p]),(Subscript[α, p])}}},{{{(Subscript[b, 1]),(Subscript[β, 1])},…,{(Subscript[b, m]),(Subscript[β, m])}},{{(Subscript[b, m+1]),(Subscript[β, m+1])},…,{(Subscript[b, q]),(Subscript[β, q])}}},z]==
(1/(2 π I))ContourIntegrate[((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])s]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) s] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) s]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) s]))(z^(-s)),{s,ℒ}]/;
m∈Integers∧m≥0∧ n∈Integers∧n≥0∧p∈Integers∧p≥0∧q∈Integers∧q≥0∧m≤q∧n≤p∧(Subscript[α, i])∈Reals∧(Subscript[α, i])>0∧1<=i<=p∧(Subscript[β, j])∈Reals∧(Subscript[β, j])>0∧1<=j<=q

In particular, if all positive coefficients (Subscript[α, i]) and (Subscript[β, j]) are integers the FoxH function can be converted to MeijerG function (which is the FoxH function with all unit coefficients  (Subscript[α, i])==(Subscript[β, j])==1):

MeijerG[{Table[(Subscript[a, i]),{i,1,n}],Table[(Subscript[a, i]),{i,n+1,p}]},{Table[(Subscript[b, i]),{i,1,m}],Table[(Subscript[b, i]),{i,m+1,q}]},z]==
MeijerG[{{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}},z]==FoxH[{Table[{(Subscript[a, i]),1},{i,1,n}],Table[{(Subscript[a, i]),1},{i,n+1,p}]},{Table[{(Subscript[b, i]),1},{i,1,m}],Table[{(Subscript[b, i]),1},{i,m+1,q}]},z]==
FoxH[{{{(Subscript[a, 1]),1},…,{(Subscript[a, n]),1}},{{(Subscript[a, n+1]),1},…,{(Subscript[a, p]),1}}},{{{(Subscript[b, 1]),1},…,{(Subscript[b, m]),1}},{{(Subscript[b, m+1]),1},…,{(Subscript[b, q]),1}}},z]

About conversion functions to FoxH & MeijerG functions to ratios of products of gamma functions

Through Meijer G and Fox H functions we can represent more than 100 named well known functions such as log, exp, arctan, Bessel, Airy and Legendre functions. In Mathematica TraditionalForm notations these functions form the following list with 127 functions to which we can apply ResourceFunction[“MeijerGForm”]:

MeijerG127={(a^z),(E^z),(Sqrt[z]),(z^b),(AiryAi[z]),(AiryAiPrime[z]),(AiryBi[z]),(AiryBiPrime[z]),(AngerJ2[a, z]),(AngerJ[a, b, z]),(cos^(-1))(z),(cosh^(-1))(z),(cot^(-1))(z),(coth^(-1))(z),(csc^(-1))(z),(csch^(-1))(z),(sec^(-1))(z),(sech^(-1))(z),(sin^(-1))(z),(sinh^(-1))(z),(tan^(-1))(z),(tan^(-1))(a,z),(tan^(-1))(z,a),(tanh^(-1))(z),
(BesselI[a, z]),(BesselJ[a, z]),(BesselK[a, z]),(BesselY[a, z]),(Beta3[z, a, b]),(Beta4[c, z, a, b]),(Beta4[z, c, a, b]),(BetaRegularized[z, a, b]),(BetaRegularized4[c, z, a, b]),(BetaRegularized4[z, c, a, b]),(CarlsonRC[x, z]),(CarlsonRC[z, y]),(CarlsonRE[x, z]),(CarlsonRE[y, z]),(CarlsonRK[x, z]),(CarlsonRK[y, z]),(ChebyshevT[a, z]),(ChebyshevU[a, z]),cos(z),cosh(z),(CoshIntegral[z]),(CosIntegral[z]),(DawsonF[z]),(EllipticE[z]),(EllipticK[z]),erf(z),erf(a,z),erf(z,b),erfc(z),erfi(z),(ExpIntegralE[a, z]),(ExpIntegralEi[z]),(Fibonacci[z]),(Fibonacci2[a, z]),(FresnelC[z]),(FresnelF[z]),(FresnelG[z]),(FresnelS[z]),(Gamma2[a, z]),(Gamma3[a, b, z]),(Gamma3[a, z, b]),(GammaRegularized[a, z]),(GammaRegularized3[a, b, z]),(GammaRegularized3[a, z, b]),(GegenbauerC[a, b, z]),(HankelH1[a, z]),(HankelH2[a, z]),hav(z),(HermiteH[a, z]),(Hypergeometric0F1[a, z]),(Hypergeometric0F1Regularized[a, z]),(Hypergeometric1F1[a, b, z]),(Hypergeometric1F1Regularized[a, b, z]),(Hypergeometric2F1[a, b, c, z]),(Hypergeometric2F1Regularized[a, b, c, z]),(Subscript[, p])(Subscript[F, q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z),(Subscript[, p])(Subscript[(OverTilde[F]), q])((Subscript[a, 1]),…,(Subscript[a, p]);(Subscript[b, 1]),…,(Subscript[b, q]);z),(HypergeometricU[a, b, z]),(hav^(-1))(z),(KelvinBei2[0, z]),(KelvinBei2[a, z]),(KelvinBer2[0, z]),(KelvinBer2[a, z]),(KelvinKei2[0, z]),(KelvinKei2[a, z]),(KelvinKer2[0, z]),(KelvinKer2[a, z]),(LaguerreL[a, z]),(LaguerreL3[a, b, z]),(LegendreP[ν, z]),(LegendreP3[a, b, z]),(LegendreP4[a, b, 2, z]),(LegendreP4[a, b, 3, z]),(LegendreQ[ν, z]),(LegendreQ3[a, b, z]),(LegendreQ4[a, b, 2, z]),(LegendreQ4[a, b, 3, z]),log(z),(LucasL[z]),(LucasL2[a, z]),(ParabolicCylinderD[a, z]),(PolyLog[2, z]),(PolyLog[a, z]),(ScorerGi[z]),(ScorerGiPrime[z]),(ScorerHi[z]),(ScorerHiPrime[z]),sin(z),sinc(z),sinh(z),(SinhIntegral[z]),(SinIntegral[z]),(SphericalBesselJ[a, z]),(SphericalBesselY[a, z]),(SphericalHankelH1[a, z]),(SphericalHankelH2[a, z]),(StruveH[ν, z]),(StruveL[ν, z]),(UnitStepSeq[z]),(WeberE2[ν, z]),(WeberE[ν, a, z]),(WhittakerM[a, b, z]),(WhittakerW[a, b, z])};

MF={#,ResourceFunction["MeijerGForm"][#,z]}&/@ MeijerG127//EchoTiming//TableForm

MR={#,MeijerGReduce[#,z]}&/@ MeijerG127//EchoTiming//TableForm

Select[#,FreeQ[#,MeijerG]&]&/@{MF[[1]],MR[[1]]}

Length[#]&/@%

Generic FoxH and G transforms and their cases

Derivation

The G transform can be defined by formula (see formula (36.23) from Samko  S . G ., Kilbas  A . A ., Marichev  O . I ., Fractional  integrals  and  derivatives : theory  and  applications, Gordon  and  Breach  Science  Publishers, 1993, 976  p . )

[f[t],t,x,{"G",{{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}}}]==(∫_0^∞)(1/t) MeijerG[{{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}},(x/t)]f[t]\[DifferentialD]t

The FoxH transform can be defined by formula (see formula (3.1.1) from ((HyperlinkTemplate[Kilbas A.A., Saigo M., H-Transforms:Theory and Applications, Chapman & Hall/CRC, 2004,389 p., https://www.taylorfrancis.com/books/mono/10.1201/9780203487372/transforms-anatoly-kilbas, https://www.taylorfrancis.com/books/mono/10.1201/9780203487372/transforms-anatoly-kilbas, HyperlinkActionRecycled, HyperlinkActive, , ])but their we find the case α==1 of below more general definition)

GenericIntegralTransform[f[t],t,x,{"H",{{{(Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1])},…,{(Subscript[(Overscript[a, -]), (Overscript[n, -])]),(Subscript[(Overscript[α, -]), (Overscript[n, -])])}},{{(Subscript[(Overscript[a, -]), (Overscript[n, -])+1]),(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])},…,{(Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])])}}},{{{(Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1])},…,{(Subscript[(Overscript[b, -]), (Overscript[m, -])]),(Subscript[(Overscript[β, -]), (Overscript[m, -])])}},{{(Subscript[(Overscript[b, -]), (Overscript[m, -])+1]),(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])},…,{(Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])])}}},α}]==(∫_0^∞)(t^(α-1))FoxH[{{{(Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1])},…,{(Subscript[(Overscript[a, -]), (Overscript[n, -])]),(Subscript[(Overscript[α, -]), (Overscript[n, -])])}},{{(Subscript[(Overscript[a, -]), (Overscript[n, -])+1]),(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])},…,{(Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])])}}},{{{(Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1])},…,{(Subscript[(Overscript[b, -]), (Overscript[m, -])]),(Subscript[(Overscript[β, -]), (Overscript[m, -])])}},{{(Subscript[(Overscript[b, -]), (Overscript[m, -])+1]),(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])},…,{(Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])])}}},t x]f[t] \[DifferentialD]t

It is easy to see, that the G transform is a particular case of the H transform:

GenericIntegralTransform[f[τ],τ,x,{"G",A,B}]==GenericIntegralTransform[f[(1/τ)],τ,x,{"H",(Overscript[A, -]),(Overscript[B, -]),0}]/;(Subscript[(Overscript[α, -]), j])==(Subscript[(Overscript[β, -]), j])==1&&(Subscript[(Overscript[a, -]), j])==(Subscript[a, j])&&(Subscript[(Overscript[b, -]), j])==(Subscript[b, j])

We can make the following sets of representations of classical integral transforms through corresponding integrals and more general G transform:

GenericIntegralTransform[f[t],t,x,"FourierCos"]==(Sqrt[(2/π)])(∫_0^∞)Cos[t x] f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"FourierSin"]==(Sqrt[(2/π)])(∫_0^∞)Sin[t x] f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"Fourier"]==(1/(Sqrt[2π]))(∫_(-∞)^∞)(E^(I t x)) f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Hankel",ν}]==(∫_0^∞)t BesselJ[ν,t x]f[t] \[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Hankel",ν,α}]==(∫_0^∞)(t^(α+1)) BesselJ[ν,t x]f[t] \[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"Hartley"]==(Sqrt[2])(∫_(-∞)^∞)Sin[t x+(π/4)] f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"Hilbert"]==(1/π)Integrate[((f(t))/(t-x)),{t,-∞,∞},PrincipalValue->True,Assumptions->x∈(Reals)]

GenericIntegralTransform[f[t],t,x,{"Hilbert",α}]==(1/π)Integrate[(t^α)((f(t))/(t-x)),{t,-∞,∞},PrincipalValue->True,Assumptions->x∈(Reals)]

GenericIntegralTransform[f[t],t,x,"Integrate"]==(∫_0^x)f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Integrate",α}]==(∫_0^x)(t^α)f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"KontorovichLebedev"]==(∫_0^∞)BesselK[I x,t]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"Laplace"]==(∫_0^∞)(E^(-t x)) f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Laplace",α}]==(∫_0^∞)(t^α) (E^(-t x)) f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"LaplaceTwoSided"]==(∫_(-∞)^∞)(E^(-t x)) f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"MehlerFock"]==(∫_0^∞)LegendreP[I t-(1/2),x]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"MehlerFock",k}]==(∫_0^∞)LegendreP[I t-(1/2),k,x]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"Meijer"]==(∫_0^∞)t BesselK[ν,t x]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Meijer",α}]==(∫_0^∞)(t^(α+1)) BesselK[ν,t x]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,"Mellin"]==(∫_0^∞)(t^(x-1)) f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Mellin",α}]==(∫_0^∞)(t^(α+x-1)) f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Neumann",ν}]==(∫_0^∞)BesselY[ν,t x]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Neumann",ν,α}]==(∫_0^∞)(t^α)BesselY[ν,t x]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Riesz",β}]==((π )/(Gamma[1-β] Sin[((π β)/2)]))(∫_0^∞)(1/((Abs[x-t])^(1-β)))f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Stieltjes",ρ}]==(∫_0^∞)((f[t])/(((x+t))^ρ))\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Struve",ν}]==(∫_0^∞)StruveH[ν,t x]f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Struve",ν,α}]==(∫_0^∞)(t^α)StruveH[ν,t x]f[t]\[DifferentialD]t  (*α-1 ?? *)

GenericIntegralTransform[f[t],t,x,{"Weyl",β}]==(∫_x^∞)((((t-x))^(β-1))/(Gamma[β]))f(t)\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Liouville",β}]==(∫_0^x)((((x-t))^(β-1))/(Gamma[β]))f(t)\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Feller",β}]==2 Gamma[β] Sin[((π β)/2)](∫_0^∞)((Sign[x-t])/((Abs[x-t])^(1-β)))f[t]\[DifferentialD]t

GenericIntegralTransform[f[t],t,x,{"Abel0",β,r}]==r Integrate[(( (t^(r-1)) ((((x^r)-(t^r)))^(β-1)))/(Gamma[β]))f[t],{t,0,x}]/;Re[β]>0&&r>0

GenericIntegralTransform[f[t],t,x,{"Abel",β,r}]==r Integrate[(((t^(r-1)) ((((t^r)-(x^r)))^(β-1)))/(Gamma[β]))f[t],{t,x,∞}]/;Re[β]>0 &&r>0

GenericIntegralTransform[f[t],t,x,{"GeneralizedLaplace",ν}]==(∫_0^∞)((f[t])/t)(E^(-(x/(2t))))ParabolicCylinderD[ν,(Sqrt[((2x)/t)])]\[DifferentialD]t/;Re[ν]<1 (* Samko(36.77)*)

GenericIntegralTransform[f[t],t,x,{"Wimp",{{ν,{(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}}}}]==(∫_0^∞)MeijerG[{{1-ν+I x,1-ν-I x,(Subscript[a, 1]),…,(Subscript[a, n])},{(Subscript[a, n+1]),…,(Subscript[a, p])}},{{(Subscript[b, 1]),…,(Subscript[b, m])},{(Subscript[b, m+1]),…,(Subscript[b, q])}},t]f[t]\[DifferentialD]t

List17={Cos[x] ,Sin[x] ,(E^(I x)),BesselJ[ν,x],Sin[x+(π/4)],(1/(1-x)),UnitStep[Abs[x]-1],(E^(-x)),BesselK[ν,x],(t^(x-1)),BesselY[ν,x],(1/((Abs[1-x])^(1-β))),(1/(((x+1))^ρ)),StruveH[ν,x],(((1-x))^(β-1))UnitStep[1-Abs[x]],(((x-1))^(β-1))UnitStep[Abs[x]-1],((Sign[x-1])/((Abs[x-1])^(1-β)))};

{#,ResourceFunction["MeijerGForm"][#,x]}&/@List17

%/.ConditionalExpression[a_. (InactiveHead[MeijerG])[u1_,u2_,v_,r_:1],Co_]:>System`MeijerGDump`SlaterForm[MeijerG[u1,u2,v,r],s]/.a_. (InactiveHead[MeijerG])[u1_,u2_,v_,r_:1]:>System`MeijerGDump`SlaterForm[MeijerG[u1,u2,v,r],s]/.ConditionalExpression[HoldPattern[Piecewise[{a_List}, b_]],c_]:>b//TableForm

Example: StieltjesTransform from FoxH

GenericIntegralTransform[Gamma[ϛ](x^(ϛ+ϝ-2))(Subscript[H, p+1,q+1]^(m,n))(a (x^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p])),(1-ϝ,σ)}, {,…,, ((Subscript[b, q]),(Subscript[β, q])),(2-ϝ-ϛ,σ)}})),x,y,{"Stieltjes",ϛ}]==(∫_0^∞)(1/(((x+y))^ϛ))(Gamma[ϛ](x^(ϛ+ϝ-2))(Subscript[H, p+1,q+1]^(m,n))(a (x^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p])),(1-ϝ,σ)}, {,…,, ((Subscript[b, q]),(Subscript[β, q])),(2-ϝ-ϛ,σ)}})))\[DifferentialD]x==(∫_0^∞)(1/(((x+y))^ϛ))(Gamma[ϛ](x^(ϛ+ϝ-2))((1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((Gamma[1-ϝ+σ t](∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t])( (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])Gamma[1-(2-ϝ-ϛ)-t σ]))(((a (x^σ)))^(-t))\[DifferentialD]t))\[DifferentialD]x==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((∫_0^∞)((Gamma[ϛ](x^(ϛ+ϝ-2)))/(((x+y))^ϛ))  (((a (x^σ)))^(-t))\[DifferentialD]x)((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((Gamma[1-ϝ+σ t](∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) ((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])Gamma[1-(2-ϝ-ϛ)-t σ]))\[DifferentialD]t==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((a^(-t)) (y^(-1-t σ+ϝ)) Gamma[1+t σ-ϝ] Gamma[-1-t σ+ϛ+ϝ])((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((Gamma[1-ϝ+σ t](∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) ((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])Gamma[1-(2-ϝ-ϛ)-t σ]))\[DifferentialD]t==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))(y^(ϝ-1))(((a (y^σ)))^(-t))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) ((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])))\[DifferentialD]t==(y^(ϝ-1))(Subscript[H, p,q]^(m,n))(a (y^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}}))/;σ>0

GenericIntegralTransform[Gamma[ϛ](x^(ϛ+ϝ-2))(Subscript[H, p+1,q+1]^(m,n))(a (x^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p])),{-1+ϛ+ϝ,-σ}}, {,…,, ((Subscript[b, q]),(Subscript[β, q])),{ϝ,-σ}}})),x,y,{"Stieltjes",ϛ}]==(∫_0^∞)(1/(((x+y))^ϛ))(Gamma[ϛ](x^(ϛ+ϝ-2))(Subscript[H, p+1,q+1]^(m,n))(a (x^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p])),{-1+ϛ+ϝ,-σ}}, {,…,, ((Subscript[b, q]),(Subscript[β, q])),{ϝ,-σ}}})))\[DifferentialD]x==(∫_0^∞)(1/(((x+y))^ϛ))(Gamma[ϛ](x^(ϛ+ϝ-2))((1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((Gamma[-1+ϛ+ϝ+t (-σ)](∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t])( (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])Gamma[1-ϝ-t (-σ)]))(((a (x^σ)))^(-t))\[DifferentialD]t))\[DifferentialD]x==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((∫_0^∞)((Gamma[ϛ](x^(ϛ+ϝ-2)))/(((x+y))^ϛ))  (((a (x^σ)))^(-t))\[DifferentialD]x)((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((Gamma[-1+ϛ+ϝ+t (-σ)](∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) ((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])Gamma[1-ϝ-t (-σ)]))\[DifferentialD]t==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((a^(-t)) (y^(-1-t σ+ϝ)) Gamma[1-ϝ-t (-σ)] Gamma[-1+ϛ+ϝ+t (-σ)])((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((Gamma[-1+ϛ+ϝ+t (-σ)](∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) ((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])Gamma[1-ϝ-t (-σ)]))\[DifferentialD]t==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))(y^(ϝ-1))(((a (y^σ)))^(-t))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) ((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])))\[DifferentialD]t==(y^(ϝ-1))(Subscript[H, p,q]^(m,n))(a (y^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}}))/;σ<0

GenericIntegralTransform[Gamma[ϛ](x^(ϛ+ϝ-2))FoxH[{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Join[Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}],{{1-ϝ,σ}}]},
{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Join[Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}],{{2-ϝ-ϛ,σ}}]},a (x^σ)],x,y,{"Stieltjes",ϛ}]==(y^(ϝ-1))FoxH[{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}]},
{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (y^σ)]/;σ>0

Inversions of generic FoxH and G transforms

Derivation

By  definition  FoxH  transform  “H” from function f[x] we  have the following integral representation:

GenericIntegralTransform[f[x],x,y,{"H",{{{(Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1])},…,{(Subscript[(Overscript[a, -]), (Overscript[n, -])]),(Subscript[(Overscript[α, -]), (Overscript[n, -])])}},{{(Subscript[(Overscript[a, -]), (Overscript[n, -])+1]),(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])},…,{(Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])])}}},{{{(Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1])},…,{(Subscript[(Overscript[b, -]), (Overscript[m, -])]),(Subscript[(Overscript[β, -]), (Overscript[m, -])])}},{{(Subscript[(Overscript[b, -]), (Overscript[m, -])+1]),(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])},…,{(Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])])}}},α}]==(∫_0^∞)(x^(α-1))f[x] FoxH[{{{(Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1])},…,{(Subscript[(Overscript[a, -]), (Overscript[n, -])]),(Subscript[(Overscript[α, -]), (Overscript[n, -])])}},{{(Subscript[(Overscript[a, -]), (Overscript[n, -])+1]),(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])},…,{(Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])])}}},{{{(Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1])},…,{(Subscript[(Overscript[b, -]), (Overscript[m, -])]),(Subscript[(Overscript[β, -]), (Overscript[m, -])])}},{{(Subscript[(Overscript[b, -]), (Overscript[m, -])+1]),(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])},…,{(Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])])}}},x y]\[DifferentialD]x

Using (OverBar[A]),(OverBar[B]) for notations corresponding lists of parameters we re-write above formula as following

GenericIntegralTransform[f[x],x,y,{"H",(OverBar[A]),(OverBar[B]),α}]==(∫_0^∞)(x^(α-1))f[x] FoxH[(OverBar[A]),(OverBar[B]),x y]\[DifferentialD]x

where

(Subscript[H, (Overscript[p, -]),(Overscript[q, -])]^((Overscript[m, -]),(Overscript[n, -])))(x y\[VerticalSeparator]({{((Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1]))}, {((Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1]))}})({{,…,, ((Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])]))}, {,…,, ((Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])]))}}))==FoxH[(OverBar[A]),(OverBar[B]),x y]

Let as apply this formula for below function f[x], which again we will use (OverBar[C]) for notations of its sets of parameters

f[x]==(Subscript[H, p+(Overscript[n, -]),q+(Overscript[m, -])]^(m+(Overscript[q, -])-(Overscript[m, -]),n+(Overscript[p, -])-(Overscript[n, -])))(a (x^σ)\[VerticalSeparator]({{{1-(Subscript[(Overscript[a, -]), (Overscript[n, -])+1])-(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])α,(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])σ},…,{1-(Subscript[(Overscript[a, -]), (Overscript[p, -])])-(Subscript[(Overscript[α, -]), (Overscript[p, -])])α,(Subscript[(Overscript[α, -]), (Overscript[p, -])])σ},{(Subscript[a, 1]),(Subscript[α, 1])}}, {{1-(Subscript[(Overscript[b, -]), (Overscript[m, -])+1])-(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])α,(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])σ},…,{1-(Subscript[(Overscript[b, -]), (Overscript[q, -])])-(Subscript[(Overscript[β, -]), (Overscript[q, -])])α,(Subscript[(Overscript[β, -]), (Overscript[q, -])])σ},{(Subscript[b, 1]),(Subscript[β, 1])}}})({{,…,{(Subscript[a, p]),(Subscript[α, p])},, {1-(Subscript[(Overscript[a, -]), 1])-(Subscript[(Overscript[α, -]), 1])α,(Subscript[(Overscript[α, -]), 1])σ},…,{1-(Subscript[(Overscript[a, -]), (Overscript[n, -])])-(Subscript[(Overscript[α, -]), (Overscript[n, -])])α,(Subscript[(Overscript[α, -]), (Overscript[n, -])])σ}}, {,…,{(Subscript[b, q]),(Subscript[β, q])},, {1-(Subscript[(Overscript[b, -]), 1])-(Subscript[(Overscript[β, -]), 1])α,(Subscript[(Overscript[β, -]), 1])σ},…,{1-(Subscript[(Overscript[b, -]), (Overscript[m, -])])-(Subscript[(Overscript[β, -]), (Overscript[m, -])])α,(Subscript[(Overscript[β, -]), (Overscript[m, -])])σ}}}))==(Subscript[H, p+(Overscript[n, -]),q+(Overscript[m, -])]^(m+(Overscript[q, -])-(Overscript[m, -]),n+(Overscript[p, -])-(Overscript[n, -])))(a (x^σ)\[VerticalSeparator](OverBar[C]))

Then we come to below chain of manipulations in which ratio of products ((((∏_(j=1)^(Overscript[m, -]))Gamma[(Subscript[(Overscript[b, -]), j])+(Subscript[(Overscript[β, -]), j])α -(Subscript[(Overscript[β, -]), j])σ t]) (∏_(i=1)^(Overscript[n, -]))Gamma[1-(Subscript[(Overscript[a, -]), i])-(Subscript[(Overscript[α, -]), i])α+(Subscript[(Overscript[α, -]), i]) σ t] )/(((∏_(i=(Overscript[n, -])+1)^(Overscript[p, -]))Gamma[(Subscript[(Overscript[a, -]), i])+(Subscript[(Overscript[α, -]), i])α-(Subscript[(Overscript[α, -]), i])σ t]) (∏_(j=(Overscript[m, -])+1)^(Overscript[q, -]))Gamma[1-(Subscript[(Overscript[b, -]), j])-(Subscript[(Overscript[β, -]), j])α+(Subscript[(Overscript[β, -]), j]) σ t])) will appear in numerator and in the denominator, that makes its cancelation and we get simple FoxH function WITHOUT parameters with tildes.

(∫_0^∞)(x^(α-1))f[x] FoxH[(OverBar[A]),(OverBar[B]),x y]\[DifferentialD]x==(∫_0^∞)(x^(α-1))(Subscript[H, p+(Overscript[n, -]),q+(Overscript[m, -])]^(m+(Overscript[q, -])-(Overscript[m, -]),n+(Overscript[p, -])-(Overscript[n, -])))(a (x^σ)\[VerticalSeparator](OverBar[C]))* FoxH[(OverBar[A]),(OverBar[B]),x y]\[DifferentialD]x==(∫_0^∞)(x^(α-1))((1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))(((∏_(j=(Overscript[m, -])+1)^(Overscript[q, -]))Gamma[1-(Subscript[(Overscript[b, -]), j])-(Subscript[(Overscript[β, -]), j])α+(Subscript[(Overscript[β, -]), j]) σ t]((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]))/((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t] (∏_(i=1)^(Overscript[n, -]))Gamma[1-(Subscript[(Overscript[a, -]), i])-(Subscript[(Overscript[α, -]), i])α+(Subscript[(Overscript[α, -]), i]) σ t] ))((((∏_(i=(Overscript[n, -])+1)^(Overscript[p, -]))Gamma[(Subscript[(Overscript[a, -]), i])+(Subscript[(Overscript[α, -]), i])α-(Subscript[(Overscript[α, -]), i])σ t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/(((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])((∏_(j=1)^(Overscript[m, -]))Gamma[(Subscript[(Overscript[b, -]), j])+(Subscript[(Overscript[β, -]), j])α -(Subscript[(Overscript[β, -]), j])σ t])))(((a (x^σ)))^(-t))\[DifferentialD]t)*
(Subscript[H, (Overscript[p, -]),(Overscript[q, -])]^((Overscript[m, -]),(Overscript[n, -])))(x y\[VerticalSeparator]({{((Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1]))}, {((Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1]))}})({{,…,, ((Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])]))}, {,…,, ((Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])]))}})) \[DifferentialD]x==
(∫_0^∞)(x^(α-1))((1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))(((∏_(j=(Overscript[m, -])+1)^(Overscript[q, -]))Gamma[1-(Subscript[(Overscript[b, -]), j])-(Subscript[(Overscript[β, -]), j])α+(Subscript[(Overscript[β, -]), j]) σ t]((∏_(i=(Overscript[n, -])+1)^(Overscript[p, -]))Gamma[(Subscript[(Overscript[a, -]), i])+(Subscript[(Overscript[α, -]), i])α-(Subscript[(Overscript[α, -]), i])σ t]))/( (∏_(i=1)^(Overscript[n, -]))Gamma[1-(Subscript[(Overscript[a, -]), i])-(Subscript[(Overscript[α, -]), i])α+(Subscript[(Overscript[α, -]), i]) σ t] ((∏_(j=1)^(Overscript[m, -]))Gamma[(Subscript[(Overscript[b, -]), j])+(Subscript[(Overscript[β, -]), j])α -(Subscript[(Overscript[β, -]), j])σ t])))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])))(((a (x^σ)))^(-t))\[DifferentialD]t)*
(Subscript[H, (Overscript[p, -]),(Overscript[q, -])]^((Overscript[m, -]),(Overscript[n, -])))(x y\[VerticalSeparator]({{((Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1]))}, {((Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1]))}})({{,…,, ((Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])]))}, {,…,, ((Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])]))}})) \[DifferentialD]x==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((∫_0^∞)(Subscript[H, (Overscript[p, -]),(Overscript[q, -])]^((Overscript[m, -]),(Overscript[n, -])))(x y\[VerticalSeparator]({{((Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1]))}, {((Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1]))}})({{,…,, ((Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])]))}, {,…,, ((Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])]))}}))(((x^(α-1))(a (x^σ)))^(-t))\[DifferentialD]x)(1/((((∏_(j=1)^(Overscript[m, -]))Gamma[(Subscript[(Overscript[b, -]), j])+(Subscript[(Overscript[β, -]), j])α -(Subscript[(Overscript[β, -]), j])σ t]) (∏_(i=1)^(Overscript[n, -]))Gamma[1-(Subscript[(Overscript[a, -]), i])-(Subscript[(Overscript[α, -]), i])α+(Subscript[(Overscript[α, -]), i]) σ t] )/(((∏_(i=(Overscript[n, -])+1)^(Overscript[p, -]))Gamma[(Subscript[(Overscript[a, -]), i])+(Subscript[(Overscript[α, -]), i])α-(Subscript[(Overscript[α, -]), i])σ t]) (∏_(j=(Overscript[m, -])+1)^(Overscript[q, -]))Gamma[1-(Subscript[(Overscript[b, -]), j])-(Subscript[(Overscript[β, -]), j])α+(Subscript[(Overscript[β, -]), j]) σ t])))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])))\[DifferentialD]t
==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((a^(-t))(y^(-α+t σ))((((∏_(j=1)^(Overscript[m, -]))Gamma[(Subscript[(Overscript[b, -]), j])+(Subscript[(Overscript[β, -]), j])α -(Subscript[(Overscript[β, -]), j])σ t]) (∏_(i=1)^(Overscript[n, -]))Gamma[1-(Subscript[(Overscript[a, -]), i])-(Subscript[(Overscript[α, -]), i])α+(Subscript[(Overscript[α, -]), i]) σ t] )/(((∏_(i=(Overscript[n, -])+1)^(Overscript[p, -]))Gamma[(Subscript[(Overscript[a, -]), i])+(Subscript[(Overscript[α, -]), i])α-(Subscript[(Overscript[α, -]), i])σ t]) (∏_(j=(Overscript[m, -])+1)^(Overscript[q, -]))Gamma[1-(Subscript[(Overscript[b, -]), j])-(Subscript[(Overscript[β, -]), j])α+(Subscript[(Overscript[β, -]), j]) σ t])))(1/((((((∏_(j=1)^(Overscript[m, -]))Gamma[(Subscript[(Overscript[b, -]), j])+(Subscript[(Overscript[β, -]), j])α -(Subscript[(Overscript[β, -]), j])σ t]) (∏_(i=1)^(Overscript[n, -]))Gamma[1-(Subscript[(Overscript[a, -]), i])-(Subscript[(Overscript[α, -]), i])α+(Subscript[(Overscript[α, -]), i]) σ t] )/(((∏_(i=(Overscript[n, -])+1)^(Overscript[p, -]))Gamma[(Subscript[(Overscript[a, -]), i])+(Subscript[(Overscript[α, -]), i])α-(Subscript[(Overscript[α, -]), i])σ t]) (∏_(j=(Overscript[m, -])+1)^(Overscript[q, -]))Gamma[1-(Subscript[(Overscript[b, -]), j])-(Subscript[(Overscript[β, -]), j])α+(Subscript[(Overscript[β, -]), j]) σ t])))))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])))\[DifferentialD]t==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((a^(-t))(y^(-α+t σ)))(1/1)((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])))\[DifferentialD]t==(y^(-α))(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])))(((a (y^(-σ))))^(-t))\[DifferentialD]t==(y^(-α))(Subscript[H, p,q]^(m,n))(a (y^(-σ))\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}}))/;σ>0

f[x]==(Subscript[H, p+(Overscript[m, -]),q+(Overscript[n, -])]^(m+(Overscript[p, -])-(Overscript[n, -]),n+(Overscript[q, -])-(Overscript[m, -])))(a (x^σ)\[VerticalSeparator]({{{(Subscript[(Overscript[b, -]), (Overscript[m, -])+1])+(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])α,-(Subscript[(Overscript[β, -]), (Overscript[m, -])+1])σ},…,{(Subscript[(Overscript[b, -]), (Overscript[q, -])])+(Subscript[(Overscript[β, -]), (Overscript[q, -])])α,-(Subscript[(Overscript[β, -]), (Overscript[q, -])])σ},{(Subscript[a, 1]),(Subscript[α, 1])}}, {{(Subscript[(Overscript[a, -]), (Overscript[n, -])+1])+(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])α,-(Subscript[(Overscript[α, -]), (Overscript[n, -])+1])σ},…,{(Subscript[(Overscript[a, -]), (Overscript[p, -])])+(Subscript[(Overscript[α, -]), (Overscript[p, -])])α,-(Subscript[(Overscript[α, -]), (Overscript[p, -])])σ},{(Subscript[b, 1]),(Subscript[β, 1])}}})({{,…,{(Subscript[a, p]),(Subscript[α, p])},, {(Subscript[(Overscript[b, -]), 1])+(Subscript[(Overscript[β, -]), 1])α,-(Subscript[(Overscript[β, -]), 1])σ},…,{(Subscript[(Overscript[b, -]), (Overscript[m, -])])+(Subscript[(Overscript[β, -]), (Overscript[m, -])])α,-(Subscript[(Overscript[β, -]), (Overscript[m, -])])σ}}, {,…,{(Subscript[b, q]),(Subscript[β, q])},, {(Subscript[(Overscript[a, -]), 1])+(Subscript[(Overscript[α, -]), 1])α,-(Subscript[(Overscript[α, -]), 1])σ},…,{(Subscript[(Overscript[a, -]), (Overscript[n, -])])+(Subscript[(Overscript[α, -]), (Overscript[n, -])])α,-(Subscript[(Overscript[α, -]), (Overscript[n, -])])σ}}}))==(Subscript[H, p+(Overscript[m, -]),q+(Overscript[n, -])]^(m+(Overscript[p, -])-(Overscript[n, -]),n+(Overscript[q, -])-(Overscript[m, -])))(a (x^σ)\[VerticalSeparator](OverBar[C]))

It means, that H-transform from initial large H-function f[x] with tildes produced FoxH-function without tildes, which again means, that INVERSE transform from right side is equal to initial large H-function f[x] with tildes.

Above evaluations were done under condition σ>0. Similar chain evaluations for σ<0 allows to write the following formula for presented above function f[x]:

(∫_0^∞)(x^(α-1))f[x] FoxH[(OverBar[A]),(OverBar[B]),x y]\[DifferentialD]x==(∫_0^∞)(x^(α-1))(Subscript[H, p+(Overscript[m, -]),q+(Overscript[n, -])]^(m+(Overscript[p, -])-(Overscript[n, -]),n+(Overscript[q, -])-(Overscript[m, -])))(a (x^σ)\[VerticalSeparator](OverBar[C]))* FoxH[(OverBar[A]),(OverBar[B]),x y]\[DifferentialD]x==((1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(i=(Overscript[n, -])+1)^(Overscript[p, -]))Gamma[(Subscript[(Overscript[a, -]), i])+(Subscript[(Overscript[α, -]), i])α+(Subscript[(Overscript[α, -]), i])(-σ) t])((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]))/((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]((∏_(j=1)^(Overscript[m, -]))Gamma[(Subscript[(Overscript[b, -]), j])+(Subscript[(Overscript[β, -]), j])α +(Subscript[(Overscript[β, -]), j])(-σ) t]) ))(((∏_(j=(Overscript[m, -])+1)^(Overscript[q, -]))Gamma[1-(Subscript[(Overscript[b, -]), j])-(Subscript[(Overscript[β, -]), j])α-(Subscript[(Overscript[β, -]), j]) (-σ) t] (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/(((∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t])(∏_(i=1)^(Overscript[n, -]))Gamma[1-(Subscript[(Overscript[a, -]), i])-(Subscript[(Overscript[α, -]), i])α-(Subscript[(Overscript[α, -]), i]) (-σ) t] ))(((a (x^σ)))^(-t))\[DifferentialD]t)*(Subscript[H, (Overscript[p, -]),(Overscript[q, -])]^((Overscript[m, -]),(Overscript[n, -])))(x y\[VerticalSeparator]({{((Subscript[(Overscript[a, -]), 1]),(Subscript[(Overscript[α, -]), 1]))}, {((Subscript[(Overscript[b, -]), 1]),(Subscript[(Overscript[β, -]), 1]))}})({{,…,, ((Subscript[(Overscript[a, -]), (Overscript[p, -])]),(Subscript[(Overscript[α, -]), (Overscript[p, -])]))}, {,…,, ((Subscript[(Overscript[b, -]), (Overscript[q, -])]),(Subscript[(Overscript[β, -]), (Overscript[q, -])]))}})) \[DifferentialD]x==(y^(-α))(Subscript[H, p,q]^(m,n))(a (y^(-σ))\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}}))/;σ<0

Example: Inverse Stieltjes Transform (cases σ>0 & σ<0)

GenericIntegralTransform[(y^(ϝ-1))FoxH[{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}]},
{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (y^σ)],y,x,{γ,"InverseStieltjes",ϛ}]==Gamma[ϛ](x^(ϛ+ϝ-2))FoxH[{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Join[Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}],{{1-ϝ,σ}}]},
{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Join[Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}],{{2-ϝ-ϛ,σ}}]},a (x^σ)]/;σ>0

GenericIntegralTransform[(y^(ϝ-1))FoxH[{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}]},
{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (y^σ)],y,x,{γ,"InverseStieltjes",ϛ}]==Gamma[ϛ](x^(ϛ+ϝ-2))FoxH[{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Join[Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}],{{-1+ϛ+ϝ,-σ}}]},
{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Join[Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}],{{ϝ,-σ}}]},a (x^σ)]/;σ<0

Additional material

Gamma function

MellinTransform[(E^(-x)),x,s]==LaplaceTransform[(x^(s-1)),x,1]==Gamma[s]

We converted exponential function to gamma function, using Mellin transform:

(f[x]==(E^(-x)))->(F[s]==Gamma[s])

{InverseMellinTransform[Gamma[s],s,x],(*(1/(2π I))(∫_(γ-i∞)^(γ+i∞))Gamma[s](x^(-s))\[DifferentialD]s *)(∑_(k=0)^∞)Residue[Gamma[s] (x^(-s)),{s,-k},Assumptions->{k∈(Integers)&&k≥0}]}

Stirling  formula for one gamma function :

(HyperlinkTemplate[Gamma[x]∝(Sqrt[2 π]) (x^(x-(1/2))) (E^(-x))/;(x->∞), http://functions.wolfram.com/06.05.06.0010.01, http://functions.wolfram.com/06.05.06.0010.01, HyperlinkActionRecycled, HyperlinkActive, , ])

{(x^x)==(E^(x Log[x])),(E^(-x)),(x^ν)}/;ν==-(1/2)

Analog of Stirling  formula for ratio of products of gamma function existing in definition of FoxH function:

((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])z])(∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i])z](Z^(-z)))/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) z])(∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) z]))∝C ((Gamma[(Subscript[B, 1])+(Subscript[Β, 1]) z]Gamma[1-(Subscript[A, 1])-(Subscript[Α, 1])z])/(Gamma[(Subscript[A, 2])+(Subscript[Α, 2]) z]Gamma[1-(Subscript[B, 2])-(Subscript[Β, 2]) z]))(W^(-z))/;(Abs[z]->∞)∧(Subscript[B, 1])==((1-m)/2)+(∑_(j=1)^m)(Subscript[b, j])∧(Subscript[Β, 1])==(∑_(j=1)^m)(Subscript[β, j])∧(Subscript[A, 1])==((1-n)/2)+(∑_(i=1)^n)(Subscript[a, i])∧(Subscript[Α, 1])==(∑_(i=1)^n)(Subscript[α, i])∧(Subscript[A, 2])==((1-p+n)/2)+(∑_(i=n+1)^p)(Subscript[a, i])∧(Subscript[Α, 2])==(∑_(i=n+1)^p)(Subscript[α, i])∧(Subscript[B, 2])==((1+m-q)/2)+(∑_(j=m+1)^q)(Subscript[b, j])∧(Subscript[Β, 2])==(∑_(j=m+1)^q)(Subscript[β, j])∧W==((((((∑_(j=1)^m)(Subscript[β, j])))^((∑_(j=1)^m)(Subscript[β, j])))((((∑_(j=m+1)^q)(Subscript[β, j])))^((∑_(j=m+1)^q)(Subscript[β, j])))(∏_(i=1)^p)(Subscript[α, i]^(Subscript[α, i])))/(((((∑_(i=1)^n)(Subscript[α, i])))^((∑_(i=1)^n)(Subscript[α, i])))((((∑_(i=n+1)^p)(Subscript[α, i])))^((∑_(i=n+1)^p)(Subscript[α, i])))(∏_(j=1)^q)((Subscript[β, j])^(Subscript[β, j]))))Z∧C==(((((2 π))^(m+n-((p+q)/2)))((((∑_(i=1)^n)(Subscript[α, i])))^((n/2)-(∑_(i=1)^n)(Subscript[a, i])))((((∑_(j=1)^m)(Subscript[β, j])))^((m/2)-(∑_(j=1)^m)(Subscript[b, j])))(∏_(j=1)^q)(Subscript[β, j]^((Subscript[b, j])-1/2)))/(((((∑_(i=n+1)^p)(Subscript[α, i])))^(((p-n)/2)-(∑_(i=n+1)^p)(Subscript[a, i])))((((∑_(j=m+1)^q)(Subscript[β, j])))^(((m-q)/2)+(∑_(j=m+1)^q)(Subscript[b, j])))(∏_(i=1)^p)(Subscript[α, i]^((Subscript[a, i])-1/2))))

Mellin convolution and Parseval equalities

MellinTransform[(∫_0^∞)f1[τ] f2[(t/τ)] (1/τ)\[DifferentialD]τ,t,s]==MellinTransform[f1[t],t,s]*MellinTransform[f2[t],t,s]==F1[s]F2[s]

(∫_0^∞)f1[τ] f2[(t/τ)](1/τ)\[DifferentialD]τ==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))MellinTransform[f1[t],t,s] MellinTransform[f2[t],t,s] (t^(-s))\[DifferentialD]s==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))F1[s]F2[s] (t^(-s))\[DifferentialD]s

MellinTransform[Integrate[(t^(δ-1))f1[z (t^β)]f2[(t^γ)],{t,0,∞}],z,s]==Integrate[Integrate[(t^(δ-1))f1[z (t^β)]f2[(t^γ)],{t,0,∞}](z^(s-1)),{z,0,∞}]==
Integrate[Integrate[f1[z (t^β)](z^(s-1)),{z,0,∞}](t^(δ-1))f2[(t^γ)],{t,0,∞}]==(1/γ)MellinTransform[f1[τ],τ,s]MellinTransform[f2[ξ],ξ,((δ-s β)/γ)-1]

MellinTransform[(∫_0^∞)(t^(δ-1)) f1[z (t^β)] f2[(t^γ)]\[DifferentialD]t,z,s]==(∫_0^∞)((∫_0^∞)(t^(δ-1)) f1[z (t^β)] f2[(t^γ)]\[DifferentialD]t) (z^(s-1))\[DifferentialD]z==(∫_0^∞)((∫_0^∞)f1[z (t^β)] (z^(s-1))\[DifferentialD]z) (t^(δ-1)) f2[(t^γ)]\[DifferentialD]t==(1/γ)MellinTransform[f1[τ],τ,s] MellinTransform[f2[ξ],ξ,((δ-s β)/γ)-1]

(∫_0^∞)(x^(ρ-1)) f1[b (x^β)] f2[a (x^σ)]\[DifferentialD]x==((a^(-(ρ/σ)))/(2 π I Abs[σ]))(∫_(η-I ∞)^(η+I ∞))F1[s]F2[((ρ-β s)/σ)]((((b/(a^(β/σ)))))^(-s))\[DifferentialD]s

(∫_0^∞)(x^(ρ-1)) f1[b x] f2[a (x^(-σ))]\[DifferentialD]x==((a^(ρ/σ))/(2 π I Abs[-σ]))(∫_(η-I ∞)^(η+I ∞))F1[s]F2[((ρ-s)/(-σ))] ((((b/(a^(-1/σ)))))^(-s))\[DifferentialD]s

LaplaceTransform from FoxH

Deriving LaplaceTransform[(x^(ρ-1))FoxH[a (x^σ)],x,s] for σ>0

LaplaceTransform[(x^(ρ-1))FoxH[a (x^σ)],x,s]==(∫_0^∞)(x^(ρ-1))FoxH[a (x^σ)] (E^(-x s))\[DifferentialD]x==(∫_0^∞)(x^(ρ-1))(Subscript[H, p,q]^(m,n))(a (x^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}})) (E^(-x s))\[DifferentialD]x==(∫_0^∞)(x^(ρ-1)) ((1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t]))(((a (x^σ)))^(-t))\[DifferentialD]t)(E^(-x s))\[DifferentialD]x==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((∫_0^∞)(E^(-s x)) (x^(-1+ρ)) (((a (x^σ)))^(-t))\[DifferentialD]x)((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t]))\[DifferentialD]t==(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))(a^(-t)) (s^(-ρ+t σ)) Gamma[ρ-t σ]((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) (∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t]))\[DifferentialD]t==(s^(-ρ))(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) ((∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t]) Gamma[ρ-σ t])/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t]))(((a (s^(-σ))))^(-t))\[DifferentialD]t==(s^(-ρ))(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) ((∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t]) Gamma[ρ-σ t] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t]))(((a (s^(-σ))))^(-t))\[DifferentialD]t==(s^(-ρ))(Subscript[H, p+1,q]^(m,n+1))(a (s^(-σ))\[VerticalSeparator]({{(1-ρ,σ),((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}}))==(s^(-ρ))FoxH[{{(1-ρ,σ),{(Subscript[a, 1]),(Subscript[α, 1])},…,{(Subscript[a, n]),(Subscript[α, n])}},{{(Subscript[a, n+1]),(Subscript[α, n+1])},…,{(Subscript[a, p]),(Subscript[α, p])}}},{{{(Subscript[b, 1]),(Subscript[β, 1])},…,{(Subscript[b, m]),(Subscript[β, m])}},{{(Subscript[b, m+1]),(Subscript[β, m+1])},…,{(Subscript[b, q]),(Subscript[β, q])}}},a (s^(-σ))]/;σ>0

result for LaplaceTransform[(x^(ρ-1))FoxH[a (x^σ)],x,s] (cases σ>0 & σ<0)

(∫_0^∞)(x^(ρ-1))(Subscript[H, p,q]^(m,n))(a (x^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}})) (E^(-x s))\[DifferentialD]x==(s^(-ρ))(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))((((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) ((∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t]) Gamma[ρ-σ t] )/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t]))(((a (s^(-σ))))^(-t))\[DifferentialD]t==(s^(-ρ))(Subscript[H, p+1,q]^(m,n+1))(a (s^(-σ))\[VerticalSeparator]({{(1-ρ,σ),((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}}))==(s^(-ρ))FoxH[{{(1-ρ,σ),{(Subscript[a, 1]),(Subscript[α, 1])},…,{(Subscript[a, n]),(Subscript[α, n])}},{{(Subscript[a, n+1]),(Subscript[α, n+1])},…,{(Subscript[a, p]),(Subscript[α, p])}}},{{{(Subscript[b, 1]),(Subscript[β, 1])},…,{(Subscript[b, m]),(Subscript[β, m])}},{{(Subscript[b, m+1]),(Subscript[β, m+1])},…,{(Subscript[b, q]),(Subscript[β, q])}}},a (s^(-σ))]/;σ>0

(∫_0^∞)(x^(ρ-1))(Subscript[H, p,q]^(m,n))(a (x^σ)\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}})) (E^(-x s))\[DifferentialD]x==(s^(-ρ))(1/(2 π I))(∫_(γ-I ∞)^(γ+I ∞))(( Gamma[ρ+(-σ)t]((∏_(j=1)^m)Gamma[(Subscript[b, j])+(Subscript[β, j])t]) ((∏_(i=1)^n)Gamma[1-(Subscript[a, i])-(Subscript[α, i]) t]))/(((∏_(i=n+1)^p)Gamma[(Subscript[a, i])+(Subscript[α, i]) t]) (∏_(j=m+1)^q)Gamma[1-(Subscript[b, j])-(Subscript[β, j]) t]))(((a (s^(-σ))))^(-t))\[DifferentialD]t==(s^(-ρ))(Subscript[H, p,q+1]^(m+1,n))(a (s^(-σ))\[VerticalSeparator]({{((Subscript[a, 1]),(Subscript[α, 1]))}, {(ρ,-σ),((Subscript[b, 1]),(Subscript[β, 1]))}})({{,…,, ((Subscript[a, p]),(Subscript[α, p]))}, {,…,, ((Subscript[b, q]),(Subscript[β, q]))}}))==(s^(-ρ))FoxH[{{{(Subscript[a, 1]),(Subscript[α, 1])},…,{(Subscript[a, n]),(Subscript[α, n])}},{{(Subscript[a, n+1]),(Subscript[α, n+1])},…,{(Subscript[a, p]),(Subscript[α, p])}}},{{(ρ,-σ),{(Subscript[b, 1]),(Subscript[β, 1])},…,{(Subscript[b, m]),(Subscript[β, m])}},{{(Subscript[b, m+1]),(Subscript[β, m+1])},…,{(Subscript[b, q]),(Subscript[β, q])}}},a (s^(-σ))]/;σ<0

InverseLaplaceTransform from FoxH for σ>0 & σ<0

Main formula

InverseLaplaceTransform[(s^(ϝ-1))Inactive[FoxH][{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}]},{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (s^σ)],s,x]==(x^(-ϝ))Inactive[FoxH][{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Join[Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}],{{1-ϝ,σ}}]},{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (x^(-σ))]

(Subscript[ℒ, s]^(-1))[(s^(ϝ-1)) (InactiveHead[FoxH])({Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}]},{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (s^σ))](x)==(x^(-ϝ)) (InactiveHead[FoxH])({Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Join[Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}],(({{1-ϝ, σ}}))]},{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (x^(-σ)))

(Subscript[ℒ, s]^(-1))[(s^(ϝ-1)) (Subscript[H, p,q]^(m,n))(a (s^σ)\[VerticalSeparator]({{{(Subscript[a, 1]),(Subscript[α, 1])}}, {{(Subscript[b, 1]),(Subscript[β, 1])}}})({{,…,, {(Subscript[a, p]),(Subscript[α, p])}}, {,…,, {(Subscript[b, q]),(Subscript[β, q])}}}))](x)==(x^(-ϝ)) (Subscript[H, p+1,q]^(m,n))(a (x^(-σ))\[VerticalSeparator]({{{(Subscript[a, 1]),(Subscript[α, 1])}}, {{(Subscript[b, 1]),(Subscript[β, 1])}}})({{,…,, {(Subscript[a, p]),(Subscript[α, p])},{({{1-ϝ,, σ}})}}, {,…,, {(Subscript[b, q]),(Subscript[β, q])}}}))

Example   (x^(ρ-1))Gamma[ζ](((1+a (x^(-σ))))^(-ζ))/;σ=±2

LaplaceTransform[With[{σ=2},(((a^(-ζ)) (x^(-1+ρ+ζ σ)) Gamma[ζ] HypergeometricPFQ[{ζ},{ζ+(ρ/2),(1/2)+ζ+(ρ/2)},-((x^σ)/(4 a))])/(Gamma[2 ζ+ρ]))],x,s]

InverseLaplaceTransform[(a^(-ζ)) (s^(-ρ)) ((((1/a)+(s^2)))^(-ζ)) Gamma[ζ],s,x]

FoxHReduce[Gamma[ζ](((1+x))^(-ζ)),x]/.x->a (x^(-σ))

Gamma[ζ](((1+a (x^(-σ))))^(-ζ))==(InactiveHead[FoxH])[{{{1-ζ,1}},{}},{{{0,1}},{}},a (x^(-σ))]

With[{n=1,p=1,m=1,q=1},{NIntegrate[(x^(ρ-1))Gamma[ζ](((1+a (x^(-σ))))^(-ζ))(E^(-x s)),{x,0,∞}],Int[(x^(ρ-1))Inactive[FoxH][{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Join[Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}],{{ρ,σ}}]},{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (x^(-σ))](E^(-x s))],(s^(-ρ))Inactive[FoxH][{Table[{(Subscript[a, i]),(Subscript[α, i])},{i,1,n}],Table[{(Subscript[a, i]),(Subscript[α, i])},{i,n+1,p}]},{Table[{(Subscript[b, i]),(Subscript[β, i])},{i,1,m}],Table[{(Subscript[b, i]),(Subscript[β, i])},{i,m+1,q}]},a (s^σ)]}]/.{(Subscript[a, 1])->1-ζ,(Subscript[α, 1])->1,(Subscript[b, 1])->0,(Subscript[β, 1])->1}//Quiet

(E^(-s x)) (x^(-1+ρ)) FoxH[{{{1-ζ,1}},{{ρ,2}}},{{{0,1}},{}},a (x^(-σ))]//FunctionExpand

With[{a=((Random[])/10),ζ=Random[],ρ=Random[],σ=2,s=Random[]},Activate[{NIntegrate[(((E^(-s x)) (x^(-1+ρ)) (((a (x^(-σ))))^(-ζ)) Gamma[ζ] HypergeometricPFQ[{ζ},{ζ+(ρ/2),(1/2)+ζ+(ρ/2)},-((x^σ)/(4 a))])/(Gamma[2 ζ+ρ])),{x,0,∞}],(s^(-ρ)) FoxH[{{{1-ζ,1}},{}},{{{0,1}},{}},a (s^σ)]}]]

(E^(-s x)) (x^(-1+ρ)) FoxH[{{{1-ζ,1}},{{ρ,-2}}},{{{0,1}},{}},a (x^(-σ))]//FunctionExpand

With[{a=((Random[])/10),ζ=Random[],ρ=Random[],σ=-2,s=Random[]},Activate[{NIntegrate[(((E^(-s x)) (x^(-1+ρ)) Gamma[ζ] HypergeometricPFQ[{ζ},{(1/2)+(ρ/2),(ρ/2)},-(1/4) a (x^(-σ))])/(Gamma[ρ])),{x,0,∞}],(s^(-ρ)) (InactiveHead[FoxH])[{{{1-ζ,1}},{}},{{{0,1}},{}},a (s^σ)]}]]

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