©2023 Ada Academica https://adac.eeEur. J. Math. Anal. 3 (2023) 6doi: 10.28924/ada/ma.3.6
Fractionalization of Hankel Type Integral Transforms and Their Relevance

B. B. Waphare∗, R. Z. Shaikh
MIT Arts, Commerce and Science College, Alandi(D), Pune, Maharashtra, India

balasahebwaphare@gmail.com, shaikhrahilanaz@gmail.com
∗Correspondence: balasahebwaphare@gmail.com

Abstract. In this paper, the fractionalization of certain types of Hankel transforms is suggested.Barut-Girardello type transforms are then introduced along with the relevant fractional order forms.Finally some further generalizations are suggested.

1. Introduction
The theory of Hankel transforms is very vast and it is studied by many researchers in recent aswell as in past. The forward and inverse transforms are completely symmetric and resemble theFourier transform, with the complex exponent as kernal being replaced by the bessel function offirst kind Jα−β of order α−β ≥−12 .The formal equivalence between the zeroth-order Hankel transform and the Abel transform fol-lowed by the Fourier transform is used as basis for developing fast algorithms for the computationof the zeroth-order Hankel transform [5]. Algorithms for the computation of the Hankel transformof integer order n > 0 have been proposed. On the basis of the general relation involving theHankel transform of integer order n > 0 and the Abel transform, whose kernel is modulated by theChebyshev polynomial of the first kind of order n, followed by the Fourier sine or cosine transformaccording to whether n is odd or even [5, 13].It has been evidenced in [18] the formal equivalence between the Hankel transform of order α−βand the Erdelyi-Kober fractional integral of order (α − β + 1

2
) followed by the Fourier cosinetransform, with both acted on function and the resulting transform being modulated by properly

α−β - dependent power functions of the inherent variables.Notably such as equivalence suggests a tool for the optical computation of Erdelyi-Kober typefractional integrals of order (n + 1
2

),n integer, through the optical implementation of the Hankeland ID Fourier transforms. In a sense, the Erdelyi-Kober type fractional integrals of order n + 1
2

Received: 4 Jun 2022.
Key words and phrases. Hankel type transform; Barut-Girardello transform; fractional transform; Erdelyi-Kobertransform. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 2
can be regarded as the (2n + 1)-plane Abe1 transform on Rm, with 0 < 2n + 1 < m.Various forms of Hankel like integral transforms have been considered in detail in a series of pa-pers [6,7,10–12,22]. We will be concerned here with certain Hankel type transform having relevanceamong others, in connection with the solution to evolution problems involving the Bessel type differ-ential operators xα+3β−1 ( ∂

∂x

)
x2(α−β)+1

(
∂
∂x

)
x−3α−β and x−3α−β ( ∂

∂x

)
x2(α−β)+1

(
∂
∂x

)
xα+3β−1with α−β ≥−1

2
and −2(α + β) real parameter [6, 11]. We introduced the fractional order formsof such Hankel-type transforms by following the lines of the fractionalization of the conventionalHankel transform [9, 16].Work of Torre [17] motivated us to prepare this paper.

2. Hankel type transforms:
The first and second Hankel-type transforms of Bessel order α−β, depending on an arbitraryreal parameter −2(α + β), respectively defined by the operational relations [6, 11].

f̃1,α,β(y) =
[
h1,α−β,−2(α+β)f

]
(y) = y1−4(α+β)

∫ ∞
0

(xy)2(α+β)Jα−β(xy)f (x)dx (1)
f̃2,α,β(y) = [h2,α−β,−2(α+β)f ](y) =

∫ ∞
0

x1−4(α+β)(xy)2(α+β)Jα−β(xy)f (x)dx (2)
where Jα−β is the Bessel type function of the first kind and order (α−β) ≥ −12 . Here f ∈ L2(R+)- the space of the complex-valued functions which are Lebesgue integrable on R+ = (0, +∞).Thetransform [h2,α−β,−2(α+β)f ](y) for α = −13 β was originally considered in [15], where the relevantcondition for its inversion were established. Also (1) and (2) relate to the Hankel type - Cliffordtransforms [10].For suitable values of α,β the Able transforms can be framed within the formalism, developedin [20, 21] concerning the integral transforms associated with complex linear transformations inquantum mechanics, which maps the position and momentum operators to canonically conjugate,but not necessarily Hermitian operator. Thus according to that formalism, the Able transformscan be seen as the radial parts of n-dimentional linear cannonical transformations, specificallyrepresenting a π/2-rotation for each pair of the cannonically conjugate operators in the respec-tive n-component position and momentum operator vectors. Precisely, n = 4(α + β) for (1) and
n = 2[1 − 2(α + β)] = 2 − 4(α + β) for (2).The order α−β of the Bessel type function relates to the eigen value λ = −l(l+n−2), l = 0, 1, 2, ...of the angular momentum; specifically, it turns out that l = α−β− n

2
+ 1, and so l = −(α+ 3β−1)for (1) and l = 3α + β for (2), thus respectively yeilding λ = 3(α2 + β2) + 10αβ − 4(α + β) + 1and λ = 3(α2 + β2) + 10αβ. When α + β = 1

4
,n = 1 in both cases, and accordingly both trans-forms yield the conventional Hankel type transform. The symmetry of (1) and (2) reflects into therelation between the respective integral kernels K1,α−β,−2(α+β)(x,y) and K2,α−β,−2(α+β)(y,x);i.e. K1,α−β,−2(α+β)(x,y) = y1−2(α+β) x2(α+β) Jα−β(xy) = K2,α−β,−2(α+β)(y,x).

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 3
As a consequence of well known orthogonality relation of the Bessel functions, both transforms (1)and (2) are self reciprocal.

h−1
1,α−β,−2(α+β) = h1,α−β,−2(α+β), h

−1
2,α−β,−2(α+β) = h2,α−β,−2(α+β). (3)

Interestingly, the adjoint operator of h1,α−β,−2(α+β) is h2,α−β,−2(α+β) and so
h∗1,α−β,−2(α+β) = h2,α−β,−2(α+β), h

∗
2,α−β,−2(α+β) = h1,α−β,−2(α+β). (4)

Also one can prove for the operator h1,α−β,−2(α+β) and h2,α−β,−2(α+β) the Parsevel equalities [12]∫ ∞
0

x−1+4(α+β) f ∗(x) g(x)dx =

∫ ∞
0

x−1+4(α+β) f̃ ∗1,α−β,−2(α+β) g̃1,α−β,−2(α+β)(x)dx,∫ ∞
0

x1−4(α+β) f ∗(x) g(x)dx =

∫ ∞
0

x1−4(α+β) f̃ ∗2,α−β,−2(α+β) g̃2,α−β,−2(α+β)(x)dx

(5)
both containing a weight function i.e. x−1+4(α+β) and x1−4(α+β) respectively. A mixed Parsevelrelation holds as well, which writes as∫ ∞

0

f ∗(x) g(x)dx =

∫ ∞
0

f̃ ∗2,α−β,−2(α+β) g̃2,α−β,−2(α+β)(x)dx. (6)
Note that it does not contain any weight function and involves both transforms [12].Relations (4) and (6) express the complementary of (1) and (2).For α + β = 1

4
, we recover the conventional Hankel type transform of Bessel order α−β;

ĥ
1,α−β,−1

2
= ĥ

2,α−β,−1
2
≡ ĥα−β with

[
ĥα−β,−2(α+β)f

]
(y) ≡ f̃α−β(y) =

∫ ∞
0

(xy)2(α+β)Jα−β(xy)f (x)dx =

∫ ∞
0

Kα,β(x,y)f (x)dx (7)
the latter expressing the transform in terms of the kernel Kα,β(x,y) = (xy)2(α+β) Jα−β(xy) =
Kα,β(y,x).Now we evidence the similarity transformations like structure of the transforms (1) and (2), beingindeed [

ĥ1,α−β,−2(α+β)f
]

(y) = (y)−2(α+β)+1/2
∫ ∞
0

(x)2(α+β)−1/2 Kα,β(x,y) f (x)dx

= (y)−2(α+β)+1/2
[
ĥα,β(x)

2(α+β)−1/2f
]

(y)[
ĥ2,α−β,−2(α+β)f

]
(y) = (y)2(α+β)−1/2

∫ ∞
0

(x)−2(α+β)+1/2 Kα,β(x,y) f (x)dx

= (y)2(α+β)−1/2
[
ĥα,β(x)

−2(α+β)+1/2f
]

(y)

(8)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 4
and so for the kernels K1,α−β,−2(α+β)(x,y) = (y)−2(α+β)+1/2 Kα,β(x,y)x2(α+β)−1/2 =
K2,α−β,−2(α+β)(y,x). It can be easily verified that [6, 11][

ĥ1,α−β,−2(α+β) B̂
∗
α−β,−2(α+β)f

]
(y) = −y2

[
ĥ1,α−β,−2(α+β)f

]
(y)[

ĥ2,α−β,−2(α+β) B̂α−β,−2(α+β)f
]

(y) = −y2
[
ĥ2,α−β,−2(α+β)f

]
(y)

(9)
where B̂α−β,−2(α+β) is the Bessel type differential operator

B̂α−β,−2(α+β) = x
α+3β−1 Dxx

2(α−β)+1 Dxx
−3α−β

= D2x + [1 − 4(α + β)]
1

x
Dx +

(3α + β)(α + 3β)

x2
(10)

whose adjoint is then
B̂∗α−β,−2(α+β) = x

−3α−β Dxx
2(α−β)+1 Dxx

α+3β−1

= D2x + [4(α + β) − 1]
1

x
Dx +

(α + 3β − 1)(3α + β − 1)
x2

(11)
Notice that for α + β = 1

4
both operators turn into the self-adjoint operator B̂α,β, being

B̂
α−β,−1

2
= B̂∗

α−β,−1
2

= D2x +
64 αβ + 3

16 x2
≡ B̂α−β. (12)

From (9) it follows, for instance, that the solution of the differential equation [6, 11].
k
∂

∂τ
h(x,τ) = B̂∗α−β,−2(α+β) (13)

satisfying the initial condition h(x, 0) = f (x) can be written into the transform conjugate y-space as
ĥ1,α−β,−2(α+β)(y,τ) = e

−y
2

kτ f̂1,α,β(y) for any value of the arbitrary constant k. Then transformingback to the x-space, one obtains
h(x,τ) =

k

2τ
x1−4(α+β)

∫ ∞
0

(xy)2(α+β)−(
k
4τ )(x

2+y2) Iα−β

(
k

2τ
xy

)
f (y)dy (14)

under the condition that |arg(τ
k

)| ≤ π
4

, which for both τ and k real turns into τ
k
> 0.

3. Hankel-type transforms of fractional order:
It is well known that equation (13) has the formal solution h(x,τ) = e τk B̂∗α,βf (x). Equation(14) yields an explicit functional representation of the exponential operator eb B̂∗α−β,−2(α+β) :[
e
b B̂∗

α−β,−2(α+β)
]
f (y) =

1

2b
y1−4(α+β)

∫ ∞
0

(xy)2(α+β) e−(
1
4b )(x

2+y2) Iα−β

(xy
2b

)
f (x)dx (15)

where Iα−β denotes the modified Bessel function of the first kind of order α − β : Jα−β(ix) =
iα−β Iα−β(x).In particular, setting b = i

2
we obtain a representation of ĥ1,α−β,−2(α+β) in the form of a symmetricfractional product of the exponential of the generators of the su(1, 1) algebra:

ĥ1,α−β,−2(α+β) = i
α−β+1e−(

i
2
)x2e

( i
2
)B̂∗

α−β,−2(α+β)e−(
i
2
)x2 (16)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 5
In this connection, we may note that the operators

K̂
(1)
+ =

1

2
x2, K̂

(1)
− = −

1

2
B̂∗α−β,−2(α+β), K̂

(1)
3 = −

i

2

(
x
d

dx
+ 2(α + β)

) (17)
conjugate a non self-adjoint one variable realization of the su(1, 1) algebra generators accordingto the inherent commutation relations[

K̂
(1)
+ ,K̂

(1)
−

]
= 2iK̂

(1)
3 ,

[
K̂
(1)
± ,K̂

(1)
3

]
= ±iK̂(1)± . (18)

Thus (16) can be formally be rewritten in terms of the operators K̂(1)+ and K̂(1)− , and further recastin the single exponential form
ĥα−β,−2(α+β) = i

α−β+1 e−i
π
2

[
K̂
(1)
+ + K̂

(1)
−

]
. (19)

This is on account of disentanglement relation for the su(1, 1) algebra generators
e
−iφ

[
K̂
(1)
+ +K̂

(1)
−

]
= e−i tan(φ/2)K̂

(1)
+ e−i sinφK̂

(1)
− e−i tan(φ/2)K̂

(1)
+ (20)

holding for −π < φ < π. Expressing (19) corresponds to the value φ = (π/2).Exploiting the integral transform representation (15) of the centred operator in equation (20), weobtain an expression for the operator e−iφ[K̂(1)+ +K̂(1)− ] in the form of a Hankel -type integral transform.Then writing φ = a(π/2) and multiplying both sides by ei(aπ/2)(α−β+1), one ends up on the L.H.S.with the ath power of the operator iα−β+1e−i π2 [K̂(1)+ +K̂(1)− ] and corresponding on the R.H.S. with
ath power of the first Hankel-type transform, Ĥa

1,α−β,−2(α+β), or the first Hankel-type transform offractional order a. Accordingly, we can write[
Ĥa1,α−β,−2(α+β)f

]
(y)

=
ei(α−β+1)(φ−π/2)

sin φ
x1−4(α+β)

∫ ∞
0

(xy)2(α+β) e
i
2
cotφ[x2+y2] Jα−β

(
xy

sin φ

)
f (x)dx (21)

= [eiφ(α−β+1) e
−iφ

[
K̂
(1)
+ +K̂

(1)
−

]
f ](y) = f̃

(a)
1,α,β

(y)

where φ = a(π/2).We can interpret it as the functional representation of the operator associated with the equation
i
∂

∂τ
h(x,τ)

= −
1

2

{
∂2

∂x2
− (1 − 4(α + β))

1

x

∂

∂x
+ [1 − 2(α + β)2 − (α−β)2]

1

x2
−x2 + 2(α−β + 1)

}
h(x,τ) (22)

with the relevant initial condition h(x, 0) = f (x).The evolution variable is here measured in unitsof (π/2) : τ = a(π/2), and conventionally denoted by φ.we can also develop similar results in relation with the second Hankel-type transform
Ĥ2,α−β,−2(α+β),the inherent su(1, 1) algebra generators being the adjoint of equation (17), namely,

K̂
(2)
+ =

1

2
x2, K̂

(2)
− = −

1

2
B̂α−β,−2(α+β), K̂

(2)
3 = −

i

2

(
x
d

dx
− 2(α + β) + 1

)
. (23)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 6
Now we introduce the second Hankel-type transform of fractional order a, ĥa

2,α−β,−2(α+β) as[
ĥa2,α−β,−2(α+β)f

]
(y)

=
ei(α−β+1)(φ−π/2)

sin φ

∫ ∞
0

x1−4(α+β) (xy)2(α+β) e
i
2
cotφ[x2+y2] Jα−β

(
xy

sin φ

)
f (x)dx (24)

=

[
eiφ(α−β+1) e

−iφ
[
K̂
(2)
+ +K̂

(2)
−

]
f

]
(y) = f̃

(a)
2,α−β,−2(α+β)(y), φ = a(π/2).

This yields the functional representation of the evolution operator for the equation
i
∂

∂τ
h(x,τ)

= −
1

2

{
∂2

∂x2
+ (1 − 2(α + β))

1

x

∂

∂x
+ [−2(α + β)2 − (α−β)2]

1

x2
−x2 + 2(α−β + 1)

}
h(x,τ) (25)

with the initial condition h(x, 0) = f (x).The evolution variable parameterized as τ = a(π/2). For
α+β = 1

4
, both equations (21) and (24) yields the expression of the conventional Hankel transform offractional order, originally introduced by Namia [9] and further investigated in [16, 21].In particular,for α + β = 1
4
, equations (17) and (23) turn into the same set of self-adjoint operators

K̂+ =
1

2
x2, K̂− = −

1

2
B̂α−β, K̂3 = −

i

2

(
x
d

dx
+

1

2

) (26)
which pertain to the conventional Hankel transform (7).Thus we have

ĥα−β = i
α−β+1e

i( π2 )
[
d2

dx2
−(α−β)2−(1/4)

x2
−x
2

2

] (27)
and corresponding for the transform of fractional order a

ĥaα−β = e
i( aπ2 )(α−β+1)e

i( aπ2 )
[
d2

dx2
−(α−β)2−(1/4)

x2
−x
2

2

] (28)
whose fractional equation is [9, 16, 21]
[
Ĥaα−βf

]
(y) ≡ f̃ (a)

α−β(y) =
ei(α−β+1)(φ−π/2)

sin φ

∫ ∞
0

(xy)(
1
2
)e

i
2
cotφ[x2+y2] Jα−β

(
xy

sin φ

)
f (x)dx (29)

since (1) and (2), also ĥa
1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) can be framed for suitable values of α,β,within the formalism of [20, 21], the relevant Canonical transformation being now the rotation bythe angle φ for each pair of corresponding canonically conjugate position and momentum operatorsin the relevant n-component operator vectors.

4. Properties of ĥa
1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β):

The fractionalization of order a of an integral transform T̂ is intended to produce an integraltransform T̂a which satisfy specific properties; more precisely we need that
(1) T̂a is continuous with respect to the order,i.e. T̂b −→ T̂a as b −→ a

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 7
(2) T̂a obeys the semigroup property, so that composing two fractional transform of order a1and a2 yeilds the fractional transform of order a1 + a2

T̂a1T̂a2 = T̂a2T̂a1 = T̂a1+a2,

(3) T̂a reduces to the identity operator with a = 0 and to the ordinary transform for a = 1; insymbols : T̂0 = 1̂ and T̂1 = T̂.
From the procedure we followed to introduced the fractional order transforms ĥa

1,α−β,−2(α+β) and
ĥa
2,α−β,−2(α+β), it is clear that both transforms satisfy the above properties.Thus the additivityproperties follows from

ĥa1,α−β,−2(α+β) =
[
ĥ1,α−β,−2(α+β)

]a
, ĥa2,α−β,−2(α+β) =

[
ĥ2,α−β,−2(α+β)

]a (30)
which in turn implies that

ĥ01,α−β,−2(α+β) = 1̂, ĥ
0
2,α−β,−2(α+β) = 1̂. (31)

In our case, the ordinary transform ĥa
1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are recovered for a = ±1:

ĥ11,α−β,−2(α+β) = ĥ1,α−β,−2(α+β), ĥ
−1
1,α−β,−2(α+β) = ĥ1,α−β,−2(α+β)

ĥ12,α−β,−2(α+β) = ĥ2,α−β,−2(α+β), ĥ
−1
2,α−β,−2(α+β) = ĥ2,α−β,−2(α+β).

(32)
This confirms to the self-reciprocal property (3) of the ordinary transform. We have

ĥ−a
1,α−β,−2(α+β) =

[
ĥa1,α−β,−2(α+β)

]−1
, ĥ−a

2,α−β,−2(α+β) =
[
ĥa2,α−β,−2(α+β)

]−1
. (33)

In fact, both transforms are periodic with respect to order parameter a i.e.
ĥ
a+2j
1,α−β,−2(α+β) = ĥ

a
1,α−β,−2(α+β), ĥ

a+2j
2,α−β,−2(α+β) = ĥ

a
2,α−β,−2(α+β) (34)

so that a can be taken in [-1,1].Intrestingly, for the adjoint operator, we find the cross-relations:[
ĥa1,α−β,−2(α+β)

]∗
= ĥ−a

2,α−β,−2(α+β),
[
ĥa2,α−β,−2(α+β)

]∗
= ĥ−a

1,α−β,−2(α+β) (35)
which reproduce (4) for a = ±1.Parsevel’s equalities separately pertaining to ĥa

1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are easily proceedi.e.∫ ∞
0

x−1+4(α+β)f (x)∗g(x)dx =

∫ ∞
0

x1+2(α+β)
[
f̃
(a)
1,α−β,−2(α+β)(x)

]∗
g̃
(a)
1,α−β,−2(α+β)(x)dx∫ ∞

0

x1−2(α+β)f (x)∗g(x)dx =

∫ ∞
0

x1−4(α+β)
[
f̃
(a)
2,α−β,−2(α+β)(x)

]∗
g̃
(a)
2,α−β,−2(α+β)(x)dx

(36)
as well as the mixed Parsevel’s relation:∫ ∞

0

f (x)∗g(x)dx =

∫ ∞
0

[
f̃
(a)
1,α−β,−2(α+β)(x)

]∗
g̃
(a)
2,α−β,−2(α+β)(x)dx. (37)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 8
For α + β = 1

4
, the above equalities turn into the energy preserving relation of the conventionaltransform: ∫ ∞

0

f (x)∗g(x)dx =

∫ ∞
0

[
f̃
(a)
α−β(x)

]∗
g̃
(a)
α−β(x)dx.

Operational rules similar to (9) can be stated for the fractional transforms, involving of courseappropriate Bessel-type differential operators. Indeed we see that[
ĥa1,α−β,−2(α+β) B̂

∗
α−β,−2(α+β),af

]
(y) = −

y2

sin2φ

[
ĥa1,α−β,−2(α+β)f

]
(y),[

ĥa2,α−β,−2(α+β) B̂
∗
α−β,−2(α+β),af

]
(y) = −

y2

sin2φ

[
ĥa2,α−β,−2(α+β)f

]
(y),

(38)
with

B̂α−β,−2(α+β),a = −2K̂
(2)
− − 2 cot

2φK̂
(2)
+ − 4 cot φK̂

(2)
3 , (39)and in particular

B̂α−β,a ≡ B̂α−β,−(1
2
),a

= B̂∗
α−β,−(1

2
),a

= −2K̂− − 2 cot2φK̂+ − 4 cot φK̂3. (40)
Relation (38) gives the relevance of the fractional transforms for the solution of differential equationsinvolving the operators B̂∗

α−β,−2(α+β),a and B̂α−β,−2(α+β),a like, for instance, the evolution equation
k
∂

∂τ
h(x,τ) = B̂∗α−β,−2(α+β),a h(x,τ), (41)

or, more in general, the following
k
∂

∂τ
h(x,τ) = P (B̂∗α−β,−2(α+β),a) h(x,τ) (42)

involving polynomial function of B̂∗
α−β,−2(α+β),a (or the adjoint involving a polynomial of

B̂α−β,−2(α+β),a).Then considering equation (41), we note that the solution turns out to be
h(x,τ) =

[
ĥ−a
1,α−β,−2(α+β) ĥ

a
α−β,−2(α+β),a(y,τ)

]
(x,τ) (43)

Before giving details of the expression of h(x,τ) from the above scheme, let us note that theoperators B̂α−β,−2(α+β),a and B̂∗α−β,−2(α+β),a arise from the adjoint transformation respectively of
B̂α−β,−2(α+β),a and B̂∗α−β,−2(α+β),a through the operator K̂(2)+ = K̂(1)+ = (x22 ). In other words:

B̂α−β,−2(α+β),a = e
−i cot(φ)

(
x2

2

)
B̂α−β,−2(α+β)e

i cot(φ)
(
x2

2

)

B̂∗α−β,−2(α+β),a = e
−i cot(φ)

(
x2

2

)
B̂∗α−β,−2(α+β)e

i cot(φ)
(
x2

2

) (44)
which can be recast as

B̂α−β,−2(α+β),a = x
α+3β−1D̂ax

2(α−β)+1D̂ax
−3α−β

B̂∗α−β,−2(α+β),a = x
−3α−βD̂ax

2(α+β)+1D̂ax
α+3β−1

(45)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 9
where D̂a being a linear combination of the Heisenberg operators x and −i ∂∂x as

D̂a = e
−i cot(φ)

(
x2

2

)
∂

∂x
e
i cot(φ)

(
x2

2

)
=

i

sin φ

[
cos(φ)x − i sin(φ)

∂

∂x

] (46)
The exponential operators ebB̂α−β,−2(α+β),a and ebB̂∗α−β,−2(α+β),a arise from the same adjoint transfor-mations of ebB̂α−β,−2(α+β) and ebB̂∗α−β,−2(α+β) respectively; viz

ebB̂α−β,−2(α+β),a = e
−i cot(φ)

(
x2

2

)
ebB̂α−β,−2(α+β)e

i cot(φ)
(
x2

2

)

e
bB̂∗

α−β,−2(α+β),a = e
−i cot(φ)

(
x2

2

)
e
bB̂∗

α−β,−2(α+β)e
i cot(φ)

(
x2

2

) (47)
which on account of (15) yields an explicit functional expression for both operators.Accordingly, thesolution of equation (41) can be written as
h(x,τ) =

k

2τ
x1−4(α+β)

∫ ∞
0

(xy)2(α+β)e−(
k
4τ
)(x2+y2)e−(

i
2
)cot(φ)(x2−y2)Iα−β

(
k

2τ
xy

)
f (y)dy (48)

under the same condition specified in connection with equation (14). One can also note thatthe similarity transformation like link between the operators B̂α−β,−2(α+β),a and B̂α−β,−2(α+β)suggests to recover equation(48) from(41) transforming h(x,τ) to h(x,τ) = h(x,τ)ei cot(φ)(x22 ).In fact the fractional transforms ĥa
1,α−β,−2(α+β) and ĥa2,α−β,−2(α+β) are linked to ĥaα−β through thesame similarity transformation holding between the ordinary transforms; viz.

ĥa1,α−β,−2(α+β) = x
−2(α+β)+1

2 ĥaα−β x
2(α+β)−1

2

ĥa2,α−β,−2(α+β) = x
2(α+β)−1

2 ĥaα−β x
−2(α+β)+1

2

as a straightforward consequence of the relations
B̂α−β,−2(α+β),a = x

2(α+β)−1
2 B̂α−β,a x

−2(α+β)+1
2

B̂∗α−β,−2(α+β),a = x
−2(α+β)+1

2 B̂α−β,a x
2(α+β)−1

2 .

5. Barut-Girardello-type transformations:
The functional expression (15) of ebB̂∗α−β,−2(α+β) resembles the Barut-Girardello-type transform.The Barut-Girardello-type transform of Bessel order α−β is defined by [3].

[
Ĝα−βf

]
(y) =

√
2

∫ ∞
0

(xy)1/2e−(1/2)(x
2+y2)Iα−β(

√
2xy)f (x)dx. (49)

As a straightforward generalization of the notion of coherent stakes associated with the Heisenbergalgebra, such generalized coherent stakes were introduced as eigenstates of the lowering operator ofthe aforementioned algebra in the relative discrete representations D±(k),k = −1/2,−1,−3/2, ...

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 10
Then taking 2b = 1√

2
in equation (15) and multiplying the integrand function by

e−(1/2)(x
2+y2)e(1/2)(x

2+y2), we end up expression[
e−(1/

√
2)K̂

(1)
− f
]

(y) = e−(y
2/2)(

√
2−1)
√

2 y1−4(α+β)
∫ ∞
0

(xy)2(α+β) e−(1/2)(x
2+y2)

Iα−β(
√

2xy) e−(x
2/2)(

√
2−1)f (x)dx.

This allows us to define the first Barut-Girardello-type transform of Bessel order α−β, dependingon a real parameter −2(α + β), through the expression[
Ĝ1,α−β,−2(α+β)f

]
(y) =

√
2 y1−4(α+β)

∫ ∞
0

(xy)2(α+β) e−(1/2)(x
2+y2) Iα−β(

√
2xy)f (x)dx. (50)

Thus we may write
Ĝ1,α−β,−2(α+β) = e

(
√
2−1)K̂(1)+ eK̂

(1)
− /
√
2 e(
√
2−1)K̂(1)+

which can eventually be imposed into the single exponential form:
Ĝ1,α−β,−2(α+β) = e

(π/4)
[
K̂
(1)
+ −K̂

(1)
−

]
. (51)

It clearly states that Ĝ1,α−β,−2(α+β) can be regarded as the evolution operator e−iτĤ, associatedwith the dynamical problem ruled by the Hamiltonian operator
Ĥ = K̂

(1)
+ − K̂

(1)
− =

1

2

[
x2 + B̂∗α−β,−2(α+β)

]
and evaluated at the purely imaginary value τ = i(π/4) of the evolution variable.Now we can define the second Barut-Girardello-type transform of Bessel order α−β as[
Ĝ2,α−β,−2(α+β)f

]
(y) =

√
2

∫ ∞
0

x1−4(α+β) (xy)2(α+β) e−(1/2)(x
2+y2)Iα−β(

√
2xy)f (x)dx (52)

for which the following operational relatoins can be stated as:
Ĝ2,α−β,−2(α+β) = e

(
√
2−1)K̂(2)+ eK̂

(2)
− /
√
2 e(
√
2−1)K̂(2)+

= e
(π/4)

[
K̂
(2)
+ −K̂

(2)
−

]
, (53)

involving of course the operators (23). Accordingly, Ĝ2,α−β,−2(α+β) can be interpreted as theevolution operator operator e−iτĤ, associated with the dynamical problem ruled by the Hamiltonianoperator
Ĥ = K̂

(2)
+ − K̂

(2)
− =

1

2

[
x2 + B̂α−β,−2(α+β)

]
and evaluated at the same complex value τ = i(π/4) of the evolution variable as Ĝ1,α−β,−2(α+β) .Both definitions (50) and(52) can be recast into the comprehensive expression[

Ĝj,α−β,−2(α+β)f
]

(y) =

∫ ∞
0

K
(BG)
j,α−β,−2(α+β)(x,y)f (x)dx, j = 1, 2. (54)

in terms of the kernels
K
(BG)
1,α−β,−2(α+β)(x,y) =

√
2y1−2(α+β)(x)2(α+β)Iα−β(

√
2xy)e−(1/2)(x

2+y2) = K
(BG)
2,α−β,−2(α+β)(y,x)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 11
which relate to the kernel K(BG)

α−β (x,y) =
√

2Iα−β(
√

2xy)e−(1/2)(x
2+y2) of the conven-tional transform (49) through the same similarity transformation like relation holding between

K1,α−β,−2(α+β)(x,y), K2,α−β,−2(α+β)(y,x) and Kα−β(x,y) .As the Hankel-type transforms, the transforms Ĝ1,α−β,−2(α+β) and Ĝ2,α−β,−2(α+β) are adjoint toeach other:
Ĝ∗1,α−β,−2(α+β) = Ĝ2,α−β,−2(α+β), Ĝ

∗
2,α−β,−2(α+β) = Ĝ1,α−β,−2(α+β). (55)

However, they are not self reciprocal; the respective inverse transforms can be easily obtained fromthe corresponding factored representation in equations (51) and (53) which yield[
Ĝ−1
1,α−β,−2(α+β)f

]
(y) = (−1)α−β+1

√
2 y1−4(α+β)

∫ ∞
0

(xy)2(α+β) e(1/2)(x
2+y2) Iα−β(

√
2xy)f (x)dx

=
{[
Ĝ−1
2,α−β,−2(α+β)

]∗
f
}

(y).

Operational relations similar to (9) can be deduced for Ĝ1,α−β,−2(α+β) and Ĝ2,α−β,−2(α+β). In fact:[
Ĝ1,α−β,−2(α+β) Î

∗
α−β,−2(α+β)f

]
(y) = 2y2

[
Ĝ1,α−β,−2(α+β)f

]
(y),[

Ĝ2,α−β,−2(α+β) Îα−β,−2(α+β)f
]

(y) = 2y2
[
Ĝ2,α−β,−2(α+β)f

]
(y),

(56)
with the differential operator Îα−β,−2(α+β) being

Îα−β,−2(α+β) = 2
[
K̂
(2)
+ − K̂

(2)
− + 2iK̂

(2)
3

]
. (57)

It can be easily seen that
Îα−β,−2(α+β) = e

−(x2/2) B̂α−β,−2(α+β) e
−(x2/2) = x2(α+β)−(α−β)−1

×
(
x +

∂

∂x

)
x2(α−β)+1

(
x +

∂

∂x

)
x−(3α+β) (58)

Even though equation (56) correspond to equation (9) , pertaining to the Hankel transform, in-volve operators Îα−β,−2(α+β) and Î∗α−β,−2(α+β) comprise also the operators K̂+ and K̂3 of thecorresponding algebras.
6. Barut-Girardello-type transforms of fractional order:

We may introduce fractional order versions of the transforms Ĝ1,α−β,−2(α+β) and Ĝ2,α−β,−2(α+β). Let us consider, the disentanglement relation for the su(1, 1) algebra generators
eζ [K̂+−K̂−] = etan(ζ/2)K̂+ e−sin(ζ)K̂− etan(ζ/2)K̂+, (59)

holding for −π < ζ < π. The expressions above obtained for Ĝ1,α−β,−2(α+β) and Ĝ2,α−β,−2(α+β)correspond to value ζ = (π/4) with appropriate set of operations (21) and (23) being respectively

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 12
involved.Let us refer in particular to the operators (17) so that on account of (15) one ends up with[

eζ [K̂
(1)
+ −K̂

(1)
− ]f

]
=

1

sin ζ
y1−4(α+β)

∫ ∞
0

(xy)2(α+β) e−(1/2)cot(ζ)(x
2+y2) Iα−β

(
xy

sin ζ

)
f (x)dx

Then,writing ζ = (aπ/4), one can obtains the ath power of (51), with the first Barut-Girardello-typetransforms of fractional order a being accordingly defined by the functional expression:[
Ĝa1,α−β,−2(α+β)f

]
(y) =

1

sin (φ/2)
y1−4(α+β)

∫ ∞
0

(xy)2(α+β) e−cot(φ/2)(x
2+y2) Iα−β

(
xy

sin (φ/2)

)
f (x)dx(60)with φ = (aπ/2), as beforeThe second Barut-Girardello-type transforms of fractional order a is similarly introduced through

e
(aπ/4)

[
K̂
(2)
+ −K̂

(2)
−

]
= Ĝa2,α−β,−2(α+β), (61)

the relevant functional expression being then:[
Ĝa2,α−β,−2(α+β)f

]
(y) =

1

sin (φ/2)

∫ ∞
0

x1−2(α+β)(xy)2(α+β)e−(1/2)cot(φ/2)(x
2+y2) Iα−β

(
xy

sin (φ/2)

)
f (x)dx.(62)The ordinary transforms are recovered, of course with a = 1, while for α + β = 1

4
, we obtainthe conventional Barut-Girardello-type transforms of fractional order a, Ĝaα−β, introduced in [16].

Ĝa
1,α−β,1/4 = Ĝ

a
2,α−β,1/4 ≡ Ĝ

a
α−β, with

[
Ĝaα−βf

]
(y) =

1

sin (φ/2)

∫ ∞
0

√
xy e−(1/2)cot(φ/2)(x

2+y2) Iα−β

(
xy

sin (φ/2)

)
f (x)dx. (63)

The fractional transforms Ĝa
1,α−β,−2(α+β) and Ĝa2,α−β,−2(α+β) are cyclic with respect to order a,being

Ĝa+8j
1,α−β,−2(α+β) = Ĝ

a
1,α−β,−2(α+β), Ĝ

a+8j
2,α−β,−2(α+β) = Ĝ

a
2,α−β,−2(α+β), (64)

which allows us to limit the values of a to the interval a ∈ [−4, 4].The operational relations (55)can be generalized to Ĝa
1,α−β,−2(α+β) and Ĝa2,α−β,−2(α+β) for which we obtain[

Ĝa1,α−β,−2(α+β) Î
∗
α−β,−2(α+β),af

]
(y) =

y2

sin2 (φ/2)

[
Ĝ1,α−β,−2(α+β)f

]
(y),

[
Ĝa2,α−β,−2(α+β) Îα−β,−2(α+β),af

]
(y) =

y2

sin2 (φ/2)

[
Ĝ2,α−β,−2(α+β)f

]
(y).

(65)
The differential operator Îα−β,−2(α+β),a is given by

Îα−β,−2(α+β),a = 2 cot
2(φ/2) K̂

(2)
+ − K̂

(2)
− + 4i cot(φ/2) K̂

(2)
3

= e[1−cot(φ/2)](x
2/2) Îα−β,−2(α+β) e

−[1−cot(φ/2)](x2/2). (66)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 13
By using equation (57) we have

Îα−β,−2(α+β),a = e
−cot(φ/2)(x2/2) B̂α−β,−2(α+β) e

cot(φ/2)(x2/2)

= e2(α+β)−(α−β)−1Îax
2(α−β)+1Îax

−2(α+β)−(α−β) (67)
with

Îa = e
−cot(φ/2)(x2/2) ∂

∂x
ecot(φ/2)(x

2/2) =
1

sin2 (φ/2)

[
cos(φ/2)x + sin(φ/2)

∂

∂x

]
. (68)

Therefore, Ĝa
1,α−β,−2(α+β) and Ĝa2,α−β,−2(α+β) are of relevance in connection with evolution equa-tions like

k
∂

∂τ
h(x,τ) = P (Î∗α−β,−2(α+β),a) h(x,τ),or

k
∂

∂τ
h(x,τ) = P (Îα−β,−2(α+β),a) h(x,τ)

involving polynomial function of Îα−β,−2(α+β),a and Î∗α−β,−2(α+β),a respectively.
7. Generalized Hankel transforms:

The H and G transform discussed above are associated with Hamiltonian operators involvinga linear combination of the generators K̂+ and K̂− of the relevant su(1, 1) algebra realizationsin a form that naturally suggests an arbitrary respectively with the attractive and repulsive radialquantum mechanics oscillator.The dynamical symmetry of the linear quantum mechanical oscillator is that of the su(1, 1) algebra,whose generator are defined in terms of the position and momentum operators are defined in terms ofthe position and momentum operators x̂ and p̂ = i ( d
dx

)
(h = 1) through the self-adjoint quadranticexpressions

K̂+ =
1

2
x̂2 =

1

2
x2, K̂− = −

i

2
p̂2 = −

1

2

d2

dx2
,

K̂3 =
1

4
(x̂p̂ + p̂x̂) = −

i

2

(
x
d

dx
+

1

2

)
.

Thus the conventional Hankel Transform of any order a, being associated with the sum operator
K̂1 = K̂+ + K̂−, turns out to be linked to the dynamics of the alternative radial oscillator, therelevant K̂− generator (12) being the radical part of the 2D Laplacian operator. In fact, as notedearlier, the Hankel transform of integer Bessel order can be regarded as the radial part of the 2DFourier transform of rotationally symmetric function, when polar co-ordinates are adopted.In otherwords we have[

F̂a f (ζ,η)
]

(x,y) = e−imφ e−imθ
[
Ĥam ρ

1/2 g(ρ)
]

(r), m = 0, 1, 2, ... (69)
where (ρ,φ) and (r,θ) are polar co-ordinates respectively in the function and transform domainand f is a rotationally symmetric function: f (ζ,η) = g(ρ) eimφ.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 14
Evidently (69) can be generalised to the transform (21) and (24) as[

F̂a f (ζ,η)
]

(x,y) = e−imφ e−imθ r−1+2(α+β)
[
Ĥa1,m ρ

1−2(α+β) g(ρ)
]

(r),[
F̂a f (ζ,η)

]
(x,y) = e−imφ e−imθ r−2(α+β)

[
Ĥa2,m,−2(α+β) ρ

2(α+β) g(ρ)
]

(r)

for non-negative integers m and rotationally symmetric functions.Likewise, the Barut-Girardello transform resorting to the difference operator K̂2 = K̂+ + K̂−,can be associated with dynamics of the repulsive radical oscillator.Therefore Barut-Girardellotransform of integer Bessel order can be regarded as the radial part of the 2D-Bargman transform
B̂a [2, 16, 20], the inherent relation being similar to (69) i.e.

[
B̂a f (ζ,η)

]
(x,y) = e−imθ r−1/2

[
Ĝam ρ

1/2 g(ρ)
]

(r), m = 0, 1, 2, ... (70)
with the same meaning of the symbols as in equation (69).The generalization of (70) to the transforms of the first and second type is obvious.Following the correspondence of the Hankel to the Fourier transform, we may introduce a gener-alized fractional Hankel transforms as the operator associated with evolution equation driven by ageneric operator belonging to the su(1, 1) algebra namely

Ĥ(1,2) = aK̂
(1,2)
+ + bK̂

(1,2)
− + cK̂

(1,2)
3 + d(v + 1)1̂

being its pertinent to the algebra realization (17) or (23).We exploit the disentanglement scheme
e−iτ[aK̂

(1,2)
+ +bK̂

(1,2)
− +cK̂

(1,2)
3 +d(v+1)1̂] = e−idτ(α−β+1) eAK̂

(1,2)
+ eCK̂

(1,2)
3 e−iφK̂

(1,2)
1 ,

giving the operator e−iτĤ(1,2) in three-term factored form, apart from the phase factor e−idτ(v+1).We may introduce the generalized Hankel-type transforms of first and second type ,depending onthe parameter P,m and γ,
[
Ĥa,P,m,γ
1,α−β,−2(α+β)f

]
(y) =

ei(α−β+1)(γ−π/2)

m sin(φ)
e−iP(y

2/2) y1−4(α+β)
∫ ∞
0

(xy)2(α+β)

×e(i/2)cot(φ)(x
2+(y2/m2)) Jα−β

(
xy

m sin (φ/2)

)
f (x)dx (71)

[
Ĥa,P,m,γ
2,α−β,−2(α+β)f

]
(y) =

ei(α−β+1)(γ−π/2)

m sin(φ)
e−iP(y

2/2)

∫ ∞
0

x1−4(α+β) (xy)2(α+β)

×e(i/2)cot(φ)(x
2+(y2/m2)) Jα−β

(
xy

m sin (φ/2)

)
f (x)dx.

The above relations reproduce (21) and (24) respectively, for b = a,d = −a,c = 0 and aτ = φ;also relations like (9) can be deduced for the generalized transform.In addition , generalized Borut-Girardello type transforms can be introduced on the basis of a

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 15
disentanglement scheme involving the operators K̂(1,2)2 instead of K̂(1,2)1 .As a conclusion, we note that from equation (71) we may recover with P = ib/m2,m2 = 1 −
b2, tan(φ) = −ib and γ = 0, the integral transformations corresponding to the exponential forms
e
bB̂∗

α−β,−2(α+β) and ebB̂α−β,−2(α+β) (see (15) and the relevant adjoint expression, easily deducible)which can respectively be regarded as the first and second Weiestrass-Gauss integral transformsof Bessel order α−β depending on the real parameter −2(α+β). They generalize to α+β = 1/4the expression of the radical Weiestrass-Gauss integral transform, for which we have [16][
Ŵα−β,bf

]
(y) =

[
e−(b/2)B̂α−βf

]
(y) =

1

b

∫ ∞
0

(xy)(1/2) e−(1/2b)(x
2+y2) Iα−β

(xy
b

)
f (x)dx

for any real parameter β > 0.
Ŵα−β,b arises as the transfer operator associated with the radial part of heat conduction likeequations.It can be in fact considered as the radial part of 2D Fresnel transform for real parametersis the optical operator for free propagation. A linear Weiestrass-Gauss integral transform has alsobeen introduced [19] the relation of Ŵm,b, m = 0, 1, 2, ... to it is evidently similar to (69), whenrationally symmetric function are involved.

Remark. [(i)](1) If we take α = ν
2
− µ
4
, β = −µ

4
− ν
2

throught this paper then all the results studied in this
paper reduce to the results studied in Torre [17].(2) Authors claim that results of this paper are stronger than that of Torre [17].

8. Conclusions:
Following the scheme already applied to other type of transforms, like for instance, the Fouriertransform , we have introduced the fractional forms of two adjoint self-reciprocal variants of theHankel type transform, which, as noted are of interest in connection with evolution problems ruledby the Bessel-type differential operators,

B̂α−β,−2(α+β) = x
α+3β−1Dxx

2(α−β)+1 Dxx
−3α−β

and
B̂α−β,−2(α+β) = x

−3α−βDxx
2(α−β)+1Dxx

α+3β−1.

The fractional order transform relate to evolution problems ruled the operators
B̂α−β,−2(α+β),a = x

α+3β−1 La
(
x,

(
∂

∂x

))
x2(α−β)+1 La

(
x,

(
∂

∂x

))
x−3α−β

and
B̂∗α−β,−2(α+β),a = x

−3α−β La
(
x,

(
∂

∂x

))
x2(α−β)+1 La

(
x,

(
∂

∂x

))
xα+3β−1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 16
where La (x,( ∂∂x)) is linear a-depending combination of x and ( ∂∂x). Since a ranges from -1 to 1,it is evident that the set of evolution problems inherent in the transforms has been greatly enlargeby the fractionalization.In general we have shown that introduced transforms can be regarded as the evolution operatorsassociated with evolution problems having an underlying su(1, 1) symmetry, the specific realizationof the algebra resorting to B̂α−β,−2(α+β) and B̂∗α−β,−2(α+β) as the relative ladder operators.Evidently, transforms of complex fractional order can be considered although of course the space offunctions on which they can meaningfully be applied must be carefully investigated. Disregardinghere this aspect of the question, we simply note that due to the additivity with respect to the order,we may write

ĤaR+ial
j,α−β,−2(α+β) = Ĥ

aR
j,α−β,−2(α+β) + Ĥ

ial
j,α−β,−2(α+β), j = 1, 2.

where aR and al respectively denote the real and imaginary part of the order a = aR +
ial .Thus ĤaRj,α−β,−2(α+β), j = 1, 2 have just the expression considered in this paper, while
Ĥial
j,α−β,−2(α+β), j = 1, 2 are from then easily deducible replacing φ with iφ.As earlier mentioned, the Hankel transform is of interest within the context of the fractional calculus[17]. Following the arguments in [18], for the transforms of our concern we find that[

Ĥ1,α−β,−2(α+β)f
]

(y) =
21−(α−β) x−α−3β+1

√
π

∫ ∞
0

cos(yτ)
[
K̂α−β+1/2g1

]
(τ)dτ,

[
Ĥ2,α−β,−2(α+β)f

]
(y) =

21−(α−β) x−α−3β+1
√
π

∫ ∞
0

cos(yτ)
[
K̂α−β+1/2g2

]
(τ)dτ

(72)
where K̂α−β+1/2 is the left hand sided Erdelyi-Kober fractional integral operator, represented by[

K̂bg
]

(y) =
1

Γ(b)

∫ ∞
y

(y2 −x2)b−1 x g(x) dx, R(b) > 0, y ∈R,

and the functions g1(x) and g2(x) on which it acts in (72) involve f as g1(x) =
xα+3β−1f (x), g2(x) = x

−3α−βf (x).Relations (72) holds under the assumption that both xg1(x) and xg2(x) are inegrable. Note thataccording to Sonine’s first integral for Bessel functions, we may say that the right hand sidedErdelyi-Kober operator of order b acts on the function xα−β Jα−β(x) as a rising operator turningit into xα−β+b Jα−β+b(x).Expressions similar to (72) can be deduced for the fractional order transforms, of course.In addition (72) as paralleled by similar expressions involving the Barut-Girardello transform ofthe first and second type. As an example, we deduce here the relation involving the conventionaltransform (49). On account of the integral representation of the modified Bessel function of thefirst kind, Iα−β, i.e.
Iα−β(xy) =

21−(α−β) yα−β x−(α−β)
√
π Γ(α−β + 1/2)

∫ x
0

(x2 −τ2)α−β−1/2 cosh(yτ)dτ, R(α−β + 1/2) > 0,

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Eur. J. Math. Anal. 10.28924/ada/ma.3.6 17
It is easy to rewrite (49) in the form

[
Ĝα−βf

]
(y) =

√
2

π
21+(

α−β
2 ) e−(1/2)y

2

∫ ∞
0

cosh(
√

2yτ) [K̂α−β+1/2h](τ)dτ,

with the function h(x) being
h(x) = x−1−(

α−β
2 ) e−(1/2)y

2

f (x).

The above relation holds under the assumption that xh(x) is integrable.We finally note that, since several forms of the Erdelyi-Kober operator exist in the literature, arather wide set of relation linking the Hankel transform to such forms can be deduced.
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	1. Introduction
	2. Hankel type transforms:
	3. Hankel-type transforms of fractional order:
	4. Properties of a1,-,-2(+) and a2,-,-2(+): 
	5. Barut-Girardello-type transformations:
	6. Barut-Girardello-type transforms of fractional order:
	7. Generalized Hankel transforms:
	8. Conclusions:
	References