Compositional Structure of Classical Integral Transforms

Oleg Marichev
Wolfram Research

ABSTRACT

The recently implemented fractional order integro-differentiation operator FractionalD is a particular case of some more general integral transforms. The majority of classical integral transforms are representable as compositions of only two transforms: the modified direct and inverse Laplace transforms with power multipliers. Using MellinTransform, one can construct two main classes of integral transforms: convolution type (FourierSinTransform, FourierExpTransform, HankelTransform, Stieltjes transform, G-transform, etc.) and transforms with respect to indices (Kontorovich–Lebedev transform, Mehler–Fock transform, Wimp transform, etc.). For these transforms, we give definitions, inversions, compositional structure, representations through more general transforms (like the G- and Wimp transforms) and more. These transforms can be directly applied to rational and other functions representable through MeijerG functions.

Introduction

About integral transforms

If somebody tries to search “integral transform”, using Google or Wikipedia or Wolfram MathWorld he will find the one-dimensional integral

(∫_a^b) f[t]K[t,x]\[DifferentialD]t==g[x]/;-∞<=a<=b<=∞

which maps function f[t] to another function g[x], using kernel function K[t,x]. This type of integrals with special kernels define such well known transformations as FourierTransform, LaplaceTransform, MellinTransform  and others. The kernels of Fourier, Laplace and Mellin transformations are the simplest elementary functions Exp and Power ((e^(a x)),(x^a)), other transforms can include special functions. Generically, from integral transforms one can select two large convolution type classes:
a) Fourier type convolution transforms with difference kernels K[t,x]==H(t-x) and
a) Mellin type convolution transforms with ratio kernels K[t,x]==G((x/t)) .
They can be transformed one to other by change of variables:

G[(y/τ)]==H[Log[(y/τ)]]==H[Log[y]-Log[τ]]==H[t-x]/;x==-Log[y]&& t==-Log[τ]&&G[ξ]==H[Log[ξ]]

This can be done effectively for transformations with a==0 or a==-∞ and b==∞.

There is exists one more large class of integral transforms: integral transforms by index of special functions in the kernels. Looking at the literature one can find just about 30 named integral transforms in these three classes. The majority of them are presented below and will be discussed in this talk were we give their definitions, formulas for inversion, some inter-relations and hierarchy. Also we show how to evaluate many of these integral transforms, using very general formulas for integrals from MeijerG functions. It is important to note that integrals here are understood in the classical sense (without involving DiracDelta and etc.).

Definitions of  integral transforms

In this section we present formulas which defined about 30 integral transforms: Abel transform, Bessel H-transform (StruveTransform), Bessel Y-transform (NeumannTransform),  Buschman transform,  (RefLink[FourierTransform, paclet:ref/FourierTransform]), (RefLink[FourierCosTransform, paclet:ref/FourierCosTransform]), (RefLink[FourierSinTransform, paclet:ref/FourierSinTransform]), Fourier-Stieltjes transform, G-transform, generalized Laplace transform,  (RefLink[HankelTransform, paclet:ref/HankelTransform]), Hartley transform,   Hermite transform, Hilbert transform, Kontorovich-Lebedev transform, (RefLink[LaplaceTransform, paclet:ref/LaplaceTransform]), Mehler–Fock transform, Generalized Mehler–Fock transform, Meijer transform, (RefLink[MellinTransform, paclet:ref/MellinTransform]), Narain G-transform, Olevski transform, (RefLink[RadonTransform, paclet:ref/RadonTransform]), Riemann-Liouville-transform ((RefLink[FractionalD, paclet:ref/FractionalD])), Riesz transform, Stieltjes transform, Two-sided Laplace transform, Weierstrass transform, W-transform and Weyl transform. Also for many of them we present their generalizations with parameter α included in factor (t^α) and give representation formulas through the more general GTransform with function MeijerG in the kernel.

Definitions

Abel transform (start at 0)

AbelTransform0==r Integrate[((f[t] (t^(r-1)) ((((x^r)-(t^r)))^(α-1)))/(Gamma[α])),{t,0,x}]/;Re[α]>0

Abel transform  (end at ∞)

AbelTransform==r Integrate[((f[t](t^(r-1)) ((((t^r)-(x^r)))^(α-1)))/(Gamma[α])),{t,x,∞}]/;Re[α]>0

Struve transform

StruveTransform[f[t],t,x,{ν,α}]==(∫_0^∞)(t^(α-1)) StruveH[ν,t x]f[t]\[DifferentialD]t

Neumann Transform

NeumannTransform[f[t],t,x,{ν,α}]==(∫_0^∞)(t^(α-1)) BesselY[ν,t x]f[t]\[DifferentialD]t

Buschman transform

(*Will be added*)

Fourier transform

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

Fourier Cos transform

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

Fourier Sin transform

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

Generalized Laplace transform

GeneralizedLaplaceTransform[f[t],t,x,ν]==(∫_0^∞)((f[t])/t)(E^(-(x/(2t))))ParabolicCylinderD[ν,(Sqrt[((2x)/t)])]\[DifferentialD]t/;Re[ν]<1

Hankel transform

HankelTransform[f[t],t,x,ν]==(∫_0^∞)t f[t] BesselJ[ν,x t]\[DifferentialD]t

Hartley transform

HartleyTransform==(∫_(-∞)^∞)f[t] Cas[x t]\[DifferentialD]t

Cas[z_]:=Cos[z]+Sin[z]==(Sqrt[2])Sin[z+(π/4)]

HartleyTransform==(Sqrt[(π/2)])FourierSinTransform[f[t]-f[-t],t,x]+(Sqrt[(π/2)])FourierCosTransform[f[t]+f[-t],t,x]

Hilbert transform

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

Integrate transform

IntegrateTransform[f[t],t,x]==(∫_0^x) f[t]\[DifferentialD]t

Kontorovich Lebedev transform

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

Laplace transform

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

Mehler Fock transform

MehlerFockTransform[f[t],t,x]==(∫_1^∞)f[t] LegendreP[I x-(1/2),t]\[DifferentialD]t

Generalized Mehler Fock transform

GeneralizedMehlerFockTransform[f[t],t,x,k]==(∫_1^∞)f[t] LegendreP[I x-(1/2),k,t]\[DifferentialD]t

Meijer transform

MeijerTransform[f[t],t,x,{ν,α}]==(∫_0^∞)(t^(α-1)) BesselK[ν,t x]f[t]\[DifferentialD]t

Mellin transform

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

Olevski transform

(*will be added*)

Riemann Liouville transform (FractionalD)

RiemannLiouvilleTransform[f[t],t,x,α]==(∫_0^x)((f[t] (((x-t))^(α-1)))/(Gamma[α]))\[DifferentialD]t  /;Re[α]>0

Riesz transform

RieszTransform[f[t], t, x,α]==(π/(Gamma[1-α] Sin[((π α)/2)]))(∫_0^∞) ((f[t])/((Abs[t-x])^(1-α)))\[DifferentialD]t/;Re[α]>0

Stieltjes transform

StieltjesTransform[f[t], t, x,ρ]==(∫_0^∞) ((f[t])/(((x+t))^ρ))\[DifferentialD]t

Two Sided Laplace transform

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

TwoSidedLaplaceTransform[f[t],t,x]==LaplaceTransform[f[t],t,x]+LaplaceTransform[f[-t],t,-x]

Weierstrass transform

WeierstrassTransform[f[t],t,x]==(1/(Sqrt[4 π]))(∫_(-∞)^∞)f[t] Exp[-(1/4) (((x-t))^2)]\[DifferentialD]t

Weyl transform

WeylTransform[f[t],t,x,α]==(∫_x^∞)((f[t] (((t-x))^(α-1)))/(Gamma[α]))\[DifferentialD]t

Definitions of  GTransform and WimpTransforms

G transform

GTransform[f[t],t,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])}}}]==(∫_0^∞)((f[t])/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)]\[DifferentialD]t

GTransform[f[t],t,x,{{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}]}}]==(∫_0^∞)((f[t])/t)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}]},(x/t)]\[DifferentialD]t

Wimp Transform

WimpTransform[f[t],t,x,{{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^∞)f[t]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]\[DifferentialD]t

WimpTransform[f[t],t,x,{{ν,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}]}}]==(∫_0^∞)f[t]MeijerG[{Table[Piecewise[{{(Subscript[a, i]),1≤i≤n},{1-ν+I x,i==n+1},{1-ν-I x,i==n+2}},0],{i,1,n+2}],Table[(Subscript[a, i]),{i,n+1,p}]},{Table[(Subscript[b, i]),{i,1,m}],Table[(Subscript[b, i]),{i,m+1,q}]},t]\[DifferentialD]t

Representation of WimpTransform through Kontorovich Lebedev and GTransforms

Wimp Transform

WimpTransform[f[t],t,x,{{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])}}}]==(2^(2-2ν))KontorovichLebedevTransform[(t^(2ν-1))GTransform[(1/τ)f[(1/τ)],τ,((t^2)/4),{{{(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,2I x]

Definitions of generalized transforms through GTransform

Abel transform (start at 0)

iAbelTransform0[{α,r},f[t],t, x]==iGTransform[{{{α},{}},{{},{0}}},(t^α) f[(t^(1/r))],t,(x^r)]

Abel transform  (end at ∞)

iAbelTransform[{α,r},f[t],t, x]==iGTransform[{{{},{α}},{{0},{}}},(t^α) f[(t^(1/r))],t,(x^r)]

Struve transform

iStruveTransform[{ν,α},f[t],t, x]==iGTransform[{{{((1+ν)/2)},{}},{{((1+ν)/2)},{-(ν/2),(ν/2)}}},(1/2) (t^(-α/2)) f[(1/(Sqrt[t]))],t,((x^2)/4)]

Neumann Transform

iNeumannTransform[{ν,α},f[t],t, x]==iGTransform[{{{},{(1/2) (-1-ν)}},{{(ν/2),-(ν/2)},{(1/2) (-1-ν)}}},(1/2) (t^(-α/2)) f[(1/(Sqrt[t]))],t,((x^2)/4)]

Buschman transform

(*Will be added*)

Fourier transform

(*Will be added*)

Fourier Cos transform

(*Will be added*)

Fourier Sin transform

(*Will be added*)

G transform

GTransform[f[t],t,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])}}}]==(∫_0^∞)((f[t])/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)]\[DifferentialD]t

Generalized Laplace transform

(*Will be added*)

Hankel transform

HankelTransform1[f[t],t,x,{ν,α}]==(∫_0^∞)(t^(α-1)) BesselJ[ν,t x]f[t]\[DifferentialD]t

iHankelTransform[{ν,α},f[t],t, x]==iGTransform[{{{},{}},{{(ν/2)},{-(ν/2)}}},(1/2) (t^(-α/2)) f[(1/(Sqrt[t]))],t,((x^2)/4)]

Hartley transform

(*Will be added*)

Hilbert transform

(*Will be added*)

Integrate transform

IntegrateTransform[f[t],t,x]==iGTransform[{{{1},{}},{{},{0}}},t f[t],t,x]

Laplace transform

iLaplaceTransform[α,f[t],t, x]==(∫_0^∞)(t^(α-1))f[t](E^(-t x))\[DifferentialD]t

iLaplaceTransform[α,f[t],t, x]==iGTransform[{{{},{}},{{0},{}}},(t^(-α)) f[(1/t)],t,x]

Mehler Fock transform

(*Will be added*)

Generalized Mehler Fock transform

(*Will be added*)

Meijer transform

MeijerTransform[f[t],t,x,{ν,α}]==(∫_0^∞)(t^(α-1)) BesselK[ν,t x]f[t]\[DifferentialD]t

iMeijerTransform[{ν,α},f[t],t, x]==iGTransform[{{{},{}},{{(ν/2),-(ν/2)},{}}},(1/4) (t^(-α/2)) f[(1/(Sqrt[t]))],t,((x^2)/4)]

Riemann Liouville transform (FractionalD)

RiemannLiouvilleTransform[f[t],t,x,α]==iGTransform[{{{α},{}},{{},{0}}},(t^α) f[t],t,x]== FractionalOrderD[f[x],{x,-α}]

Riesz transform

RieszTransform[f[t], t, x,α]==(π/(Gamma[1-α] Sin[((π α)/2)]))(∫_0^∞) ((f[t])/((Abs[t-x])^(1-α)))\[DifferentialD]t/;Re[α]>0

iRieszTransform[α,f[t],t, x]==((π Csc[((π α)/2)] )/(Gamma[1-α]))iGTransform[{{{α},{(α/2)}},{{0},{(α/2)}}},(t^α) f[t],t,x]

Stieltjes transform

StieltjesTransform[f[t], t, x,ρ]==(∫_0^∞) ((f[t])/(((x+t))^ρ))\[DifferentialD]t

iStieltjesTransform[ρ,f[t],t, x]==iGTransform[{{{1-ρ},{}},{{0},{}}},(((t^(-ρ)) f[t])/(Gamma[ρ])),t,x]

Weyl transform

WeylTransform[f[t],t,x,α]==(∫_x^∞)((f[t] (((t-x))^(α-1)))/(Gamma[α]))\[DifferentialD]t

iWeylTransform[α,f[t],t, x]==(varOut^(-1+α)) iGTransform[{{{},{α}},{{0},{}}},(t^α) f[t],t,x]

Structure of  integral transforms

Some useful books

Recently we published  (HyperlinkURL["Handbook of Mellin transforms (2018)", https://amoeba.wolfram.com/index.php/s/5bfyCpBQNPbcg4T])with a large collection formulas. Our other set of handbooks  (HyperlinkTemplate[Integrals and Series, Vols 1-5, https://www.amazon.com/Integrals-B-Prudnikov/dp/2881248373, https://www.amazon.com/Integrals-B-Prudnikov/dp/2881248373, HyperlinkActionRecycled, HyperlinkActive, , ]) is essentially more informative than the well known handbook Table of integrals, series and products by Gradshteyn and Ryzhik. Our monograph  (HyperlinkURL["Fractional Integrals and Derivatives(1993)", https://amoeba.wolfram.com/index.php/s/XH76drx5SNkdMbX]) includes detailed investigation of composition structure of integral transforms of convolution and non-convolution type (G-transform, Wimp transform and their cases). Algorithm evaluation of integrals and integral transforms was developed in (HyperlinkURL["Handbook of Integral Transforms of Higher Transcendental Functions (1983)", https://amoeba.wolfram.com/index.php/s/SzdNyiFkFGR22gq])where users can find tables of MeijerG function for building values of corresponding integrals.

Equivalence of Fourier and Mellin transforms

Looking at definitions of integral transforms one can mention, that Fourier,  Mellin and Laplace transforms have the simplest kernels: (e^(i t x)), (t^(x-1)), (e^(-t x)), which are convertible between one another by changing variables. Indeed, we have relations

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

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

TwoSidedLaplaceTransform[f[t],t,x]==(Sqrt[2π])FourierTransform[f[t],t,I x]

Fourier transform is well known, commonly covered in coursework, and has remarkable convolution property

FourierTransform[(∫_(-∞)^∞)f[t-τ]g[τ]\[DifferentialD]τ,t,z]==(Sqrt[2 π])FourierTransform[f[t],t,z]FourierTransform[g[t],t,z]

We see a similar convolution property in the Mellin transform:

MellinTransform[(∫_0^∞)(1/τ)f[(t/τ)]g[τ]\[DifferentialD]τ,t,z]==MellinTransform[f[t],t,z]MellinTransform[g[t],t,z]

But the Mellin transform is more convenient for the evaluation of integrals, using Parseval’s relation and the formula for inverse Mellin transform. In most generic format such equality for evaluation of integrals can be written as the following:

(∫_0^∞)(τ^a) f[b (τ^c)] g[d (τ^e)]\[DifferentialD]τ==((d^(-((1+a)/e)))/( Abs[e]))(1/(2π I))(∫_(γ-I ∞)^(γ+I ∞))MellinTransform[f[t],t,s] MellinTransform[g[τ],τ,((1+a-c s)/e)](((b (d^(-(c/e)))))^(-s))\[DifferentialD]s/;c≠0∧c∈Reals∧e∈Reals&&Arg[d]!=π

Many integral transforms can be represented through general GTransform with MeijerG function at the kernel

GTransform[f[t],t,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])}}}]==(∫_0^∞)((f[t])/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)]\[DifferentialD]t

For example we have

LaplaceTransform[f[t],t, x]==GTransform[{{{},{}},{{0},{}}},(1/t) f[(1/t)],t,x]

HankelTransform[f[t],t,x,ν]==(∫_0^∞)t f[t] BesselJ[ν,x t]\[DifferentialD]t==GTransform[{{{},{}},{{(ν/2)},{-(ν/2)}}}, (1/(2t)) f[(1/(Sqrt[t]))],t,((x^2)/4)]

The representations of other transforms through GTransform one can see at the top of the subsection ‘Definitions of generalized transforms through GTransform”.

In the next section we will describe the function MeijerG and how to operate with it.

Operations with MeijerG

MeijerG function is defined by formula  (see Wolfram Functions Site && MeijerG && MeijerG1 )

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]==(1/(2 π I))ContourIntegrate[((((∏_(k=1)^m)Gamma[(Subscript[b, k])+s]) (∏_(k=1)^n)Gamma[1-(Subscript[a, k])-s] )/(((∏_(k=n+1)^p)Gamma[(Subscript[a, k])+s]) (∏_(k=m+1)^q)Gamma[1-(Subscript[b, k])-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[G, p,q]^(m,n))(z\[VerticalSeparator]({{(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])}}))==(1/(2 π I))(∫_ℒ)((((∏_(k=1)^m)Γ(s+(Subscript[b, k]))) (∏_(k=1)^n)Γ(1-(Subscript[a, k])-s))/(((∏_(k=n+1)^p)Γ(s+(Subscript[a, k])) )(∏_(k=m+1)^q)Γ(1-(Subscript[b, k])-s)))(z^(-s))\[DifferentialD]s/;m∈ℕ∧n∈ℕ∧p∈ℕ∧q∈ℕ∧m≤q∧n≤p

We can represent through the MeijerG function a majority (more than 120) named functions. For this purpose one can use command MeijerGReduce or ResourceFunction[“MeijerGForm”], for example

ResourceUpdate["MeijerGForm"]

ResourceFunction["MeijerGForm"][(((1+z))^b),z]

(((InactiveHead[MeijerG])[{{1+b},{}},{{0},{}},z])/(Gamma[-b]))//Activate

MeijerGReduce[Sin[z],z]

%//Activate

Let us look at an example:

MeijerG[{{a},{}},{{c},{}},z,1]

For convenience of users, Mathematica has three useful internal commands with System`MeijerGDump` for presenting information about MeijerG’s {m,n,p,q},  gamma functions from inside of MeijerG and presentations series expansions near zero and infinity.

System`MeijerGDump`MeijerGInfo[MeijerG[{{a},{}},{{c},{}},z,1]]

System`MeijerGDump`SlaterForm[MeijerG[{{a},{}},{{c},{}},z,1],s]

System`MeijerGDump`MeijerGToSums[MeijerG[{{a},{}},{{c},{}},z,1],k]

Inactive[Sum][Residue[(z^(-s)) Gamma[1-a-s] Gamma[c+s],{s,-c-k},Assumptions->{k∈(Integers)&&k≥0}],{k,0,∞}]

%//Activate

(∑_(k=0)^∞)Residue[(z^(-s)) Gamma[1-a-s] Gamma[c+s],{s,1-a+k},Assumptions->{k∈(Integers)&&k≥0}]

Examples of compositional structure of Riemann Liouville and Hankel transforms

In below two examples we can see, that LiouvilleTransform and HankelTransform can be represented as compositions of inverse Laplace transforms with multiplicator and direct Laplace transforms. For this purpose we apply Laplace transform to corresponding transforms  LiouvilleTransform[g[τ],τ,t,α]  and HankelTransform[g[τ],τ,(Sqrt[t]),ν] and evaluate them (with parameter β==(ν/2) at lasr case):

f[t]==LiouvilleTransform[g[τ],τ,t,α]==(∫_0^t)((g[τ] (((t-τ))^(α-1)))/(Gamma[α]))\[DifferentialD]τ

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

LaplaceTransform[(t^β)f[t],t,x]==LaplaceTransform[(t^β)LiouvilleTransform[g[τ],τ,t,α],t,x]==LaplaceTransform[(t^β)(∫_0^t)((g[τ] (((t-τ))^(α-1)))/(Gamma[α]))\[DifferentialD]τ,t,x]==(∫_0^∞)(t^β)((∫_0^t)((g[τ] (((t-τ))^(α-1)))/(Gamma[α]))\[DifferentialD]τ)(E^(-t x))\[DifferentialD]t==(∫_0^∞)(∫_τ^∞)(((t^β)g[τ] (((t-τ))^(α-1))(E^(-t x)))/(Gamma[α]))\[DifferentialD]t \[DifferentialD]τ==

(∫_0^∞)((g[τ])/(Gamma[α]))((∫_τ^∞) (t^β)(((t-τ))^(α-1))(E^(-t x))\[DifferentialD]t )\[DifferentialD]τ==(∫_0^∞)((g[τ])/(Gamma[α]))(((E^(-x ξ-x τ)) (τ^(α+β)) Gamma[α] Gamma[-α-β])/(Gamma[-β]))\[DifferentialD]τ==(((E^(-x ξ)) Gamma[-α-β])/(Gamma[-β]))(∫_0^∞)(E^(-x τ)) (τ^(α+β)) g[τ]\[DifferentialD]τ==(((E^(-x ξ)) Gamma[-α-β])/(Gamma[-β]))LaplaceTransform[(τ^(α+β)) g[τ],τ,x]

f[t]==HankelTransform[g[τ],τ,(Sqrt[t]),ν]==(∫_0^∞)g[τ]τ BesselJ[ν,τ (Sqrt[t])]\[DifferentialD]τ

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

LaplaceTransform[(t^β)f[t],t,x]==LaplaceTransform[(t^β)HankelTransform[g[τ],τ,(Sqrt[t]),ν],t,x]==LaplaceTransform[(t^β)(∫_0^∞)g[τ]τ BesselJ[ν,τ (Sqrt[t])]\[DifferentialD]τ,t,x]==(∫_0^∞)(t^β)((∫_0^∞)g[τ]τ BesselJ[ν,τ (Sqrt[t])]\[DifferentialD]τ)(E^(-t x))\[DifferentialD]t==(∫_0^∞)(∫_0^∞)(t^β)g[τ] τ BesselJ[ν,τ (Sqrt[t])](E^(-t x))\[DifferentialD]t \[DifferentialD]τ==

(∫_0^∞)g[τ]τ((∫_0^∞) (t^β)BesselJ[ν,τ (Sqrt[t])](E^(-t x))\[DifferentialD]t )\[DifferentialD]τ==(∫_0^∞)g[τ]τ(2^(-ν)) (x^(-1-β-(ν/2))) (τ^ν) Gamma[1+β+(ν/2)] Hypergeometric1F1Regularized[1+β+(ν/2),1+ν,-((τ^2)/(4 x))]\[DifferentialD]τ==(∫_0^∞)g[τ]τ(2^(-ν)) (E^(-((τ^2)/(4 x)))) (x^(-1-ν)) (τ^ν)\[DifferentialD]τ==(∫_0^∞)(2^(-1-ν)) (E^(-(ξ/(4 x)))) (x^(-1-ν)) (ξ^(ν/2)) g[(Sqrt[ξ])]\[DifferentialD]ξ==(2^(-1-ν))(x^(-1-ν)) (∫_0^∞) (E^(-(ξ/(4 x)))) (ξ^(ν/2)) g[(Sqrt[ξ])]\[DifferentialD]ξ==(2^(-1-ν))(x^(-1-ν)) LaplaceTransform[(ξ^(ν/2)) g[(Sqrt[ξ])],ξ,(1/(4x))]/;β==(ν/2)

g[τ]τ(2^(-ν)) (x^(-1-β-(ν/2))) (τ^ν) Gamma[1+β+(ν/2)] Hypergeometric1F1Regularized[1+β+(ν/2),1+ν,-((τ^2)/(4 x))]/.β->ν/2

Example usage for “Neutrino forces”

Recently we found article  https://arxiv.org/abs/2209.07082v1  about “Neutrino forces in neutrino backgrounds”, which on the page 14 mentioned that Mathematica can not evaluate one integral with the Struve H function. In reality this integral after rather complicated procedures can be converted to value of StruveTransform from case of MeijerG function. It allows one to apply algorithm developed in my book of 1978 (translated in 1983, see https://en.wikipedia.org/wiki/Oleg_Marichev). As result we got formula

(∫_0^1)StruveH[ν,(Sqrt[1-(z^2)]) z]\[DifferentialD]z==((2 )/(3 π))HypergeometricPFQ[{1,1},{(5/4),(3/2),(7/4)},-(1/16)]

Evaluation of  integral transforms and integrals from MeijerG function

GTransform and its cases

GTransform

ResourceFunction["GenericIntegralTransform"][BesselY[ν,z], z, t, {"G", {{{},{α}},{{0},{}}}}, "FoxHForm"->True]

ResourceFunction["GenericIntegralTransform"][((AiryAi[z])^2), z, t, {"G", {{{},{α}},{{0},{}}}}, "FoxHForm"->True]

ResourceFunction["GenericIntegralTransform"][BesselJ[ν,(z^(1/4))] BesselK[ν,(z^(1/4))], z, t, {"G", {{{},{α}},{{0},{}}}}]

ResourceFunction["GenericIntegralTransform"][-1+((Cosh[z])^6), z, t, {"G", {{{},{α}},{{0},{}}}}]

We can apply GTransform with kernel  {“G”,{{{},{α}},{{0},{}}}}, which corresponds to

MeijerG[{{},{α}},{{0},{}},z]//FunctionExpand

As result we get

ResourceFunction["GenericIntegralTransform"][(((1+z))^(-1-(ν/2))) ChebyshevU[ν,((1+2 z)/(2 (Sqrt[z]) (Sqrt[1+z])))],z,t,{"G",{{{},{α}},{{0},{}}}}]

Hankel

GenericIntegralTransform[ArcTan[z],z,t,{"Hankel",{α,ν}}]

GenericIntegralTransform[ArcTan[z],z,t,{"Hankel",{α,ν}},"FoxHForm"->True]

Hilbert

ResourceFunction["GenericIntegralTransform"][Cos[z], z, t, {"Hilbert", 0}]

ResourceFunction["GenericIntegralTransform"][(((1+z))^(-1)), z, t, {"Hilbert", 0},"FoxHForm"->True]

Stieltjes

ResourceFunction["GenericIntegralTransform"][Sin[z],z,t,{"Stieltjes",ρ}]

ResourceFunction["GenericIntegralTransform"][BesselK[ν,z],z,t,{"Stieltjes",ρ}, "FoxHForm"->True]