International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 1 (2016), 17-23
http://www.etamaths.com

HARMONIC ANALYSIS ASSOCIATED WITH THE GENERALIZED q-BESSEL
OPERATOR

AHMED ABOUELAZ, RADOUAN DAHER, EL MEHDI LOUALID∗

Abstract. In this article, we give a new harmonic analysis associated with the generalized q-Bessel
operator. We introduce the generalized q-Bessel transform, the generalized q-Bessel translation and
the generalized q-Bessel convolution product.

1. Introduction

In this paper we consider a generalized q-Bessel operator ∆q,α,n defined by

(1) ∆q,α,nf(x) =
1

x2
[
q2nf(q−1x) − (1 + q2α+4n)f(x) + q2α+2nf(qx)

]
where n = 0, 1, ... . For n = 0, we regain the q-Bessel operator

(2) ∆q,αf(x) =
1

x2
[
f(q−1x) − (1 + q2α)f(x) + q2αf(qx)

]
Through this paper, we provide a new harmonic analysis corresponding to the generalized q-Bessel
operator ∆d,α,n.
The structure of the paper is as follows: In section 2, we set some notations and collect some basic
results about q-harmonnic analysis. In section 3, we give some facts about harmonic analysis related
to the generalized q-Bessel operator ∆q,α,n, we define the generalized q-Bessel transform and we give
some proprieties. In section 4, we define the generalized q-Bessel translation Tαq,x,n and the generalized
q-Bessel convolution product related to Tαq,x,n.

2. Element of q-harmonnic analysis

In this section, we recapitulate some facts about harmonic analysis related to the Bessel operator
∆d,α,n. We cite here, as briefly as possible, some properties. For more details we refer to [5, 6, 1, 2].
Throught this paper, we assume that 0 < q < 1 and α > −1.
let a ∈ C, the q-shifted factorial are defined by

(a; q)0 = 1, (a; q)n =

n−1∏
k=0

(1 −aqk), (a; q)∞ =
∞∏
k=0

(1 −aqk)

The q-derivative of a function f is given by

Dqf(x) =
f(x) −f(qx)

(1 −q)x
if x 6= 0

The q-Jackson integrals from 0 to a and from 0 to ∞ are defined by [3, 4]∫ a
0

f(x)dqx = (1 −q)a
∞∑
0

f(aqn)qn,

2010 Mathematics Subject Classification. 05A30, 44A15.
Key words and phrases. Generalized q-Bessel operator; Generalized q-Bessel transform; Generalized q-Bessel trans-

lation; Genaralized q-Bessel translation.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

17



18 ABOUELAZ, DAHER AND LOUALID

∫ ∞
0

f(x)dqx = (1 −q)a
∞∑

n=−∞
f(qn)qn.

We have

Dq

∫ a
x

H(t)dqt = −H(x).

The q-analogue of the integration theorem by a change of variable can be stated as follows∫ b
a

H(
λ

r
)λ2α+1dqλ = r

2α+2

∫ b
r

a
r

H(t)t2α+1dqt, ∀r ∈ R+q

The q-integration by parts formula is given by∫ b
a

g(x)Dqf(x)dqx = [f(b)g(b) −f(a)g(a)] −
∫ b
a

f(qx)Dqg(x)dq.

The third Jackson q-Bessel function Jα (also called Hahn-Exton q-Bessel functions) is defined by the
power series [7]

Jα(x; q) =
(qα+1; q)∞

(q; q)∞
xα

∞∑
n=0

(−1)n
q
n(n+1)

2

(qα+1; q)n(q; q)n
x2n,

and has the normalized form

jα(x; q) =

∞∑
n=0

(−1)n
q
n(n+1)

2

(qα+1; q)n(q; q)n
x2n.

It satisfies the following estimate [5]

|jα(qn,q2)| ≤
(−q2; q2)∞(−q2α+2; q2)∞

(q2α+2; q2)∞

{
1 if n ≥ 0
qn

2−(2α+1)n if n < 0

If x ∈ C∗ \R then we have the following asymptotic expansion as |x|→∞

jα(x; q
2) ∼

(x2q2; q2)∞
(q2α+2; q2)∞

Also the normalized q-Bessel functions satisfies an orthogonality relation

c2q,α

∫ ∞
0

jα(xt,q
2)jα(yt,q

2)t2α+1dqt = δq(x,y)

where

δq(x,y) =

{
0, if y 6= x;

1
(1−q)x2(α+1) , if x = y.

(3) cq,α =
1

1 −q
(q2α+2; q2)∞

(q2; q2)∞
.

The function x 7→ jα(λx,q2) is a solution of the following q-differential equation

(4) ∆q,αf(x) = −λ2f(x),

where ∆q,α is the q-Bessel operator given by (2).
For 1 ≤ p < ∞ we denote by Lpq,α the set of all real functions on R+q for which

‖f‖q,p,α =
(∫ ∞

0

|f(x)|px2α+1dqx
) 1
P

< ∞.

Proposition 2.1. Let f,g ∈L2q,α such that ∆q,αf, ∆q,αg ∈L2q,α then

(5)
∫ ∞
0

∆q,αf(x)g(x)x
2α+1dqx =

∫ ∞
0

f(x)∆q,αg(x)x
2α+1dqx.



GENERALIZED q-BESSEL OPERATOR 19

The q-Bessel Fourier transform Fq,α was introduced and studies in [5, 6]

(6) Fq,αf(x) = cq,α
∫ ∞
0

f(t)jα(xt,q
2)t2α+1dqt,

The q-Bessel translation operator is defined as follows [5, 6]

(7) Tαq,xf(y) = cq,α
∫ ∞
0

Fq,α(f)(t)jα(xt,q2)jα(yt,q2)t2α+1dqt.

Proposition 2.2. We have for all x,y ∈ R+q
Tαq,xf(y) = T

α
q,yf(x)

The q-translation operator is positive if

Tαq,xf ≥ 0, ∀f ≥ 0, ∀x ∈ R
+
q .

The domaine of positivity of the q-translation operator is

Qα = {q ∈]0, 1[, Tαq,x is positive for all x ∈ R
+
q }.

In [1] it was proved that if −1 < α < α′ then Qα ⊂ Qα′. As a consequence:
• if 0 ≤ α then Qα =]0, 1[.
• if −1

2
≤ α < 0 then ]0,q0[⊂ Q−1

2
⊂ Qα (]0, 1[, q0 ' 0.43.

• if −1 ≤ α < −1
2
then Qα ⊂ Q−1

2
.

In the rest of this section we always assume that the q-translation operator is positive.

The q-convolution product of two functions is given by [5, 6]

(8) f ∗q,α g(x) = cq,α
∫ ∞
0

Tαq,xf(y)g(y)y
2α+1dqy.

The following Theorem summarize some result about q-Bessel Fourier transform [6]

Theorem 2.3. The q-Bessel Fourier transform satisfies
(1) For all functions f ∈Lpq,α,

F2q,αf(x) = f(x), ∀x ∈ R
+
q .

(2) For all functions f ∈L2q,α,

(9) ‖F2q,αf ‖q,α,2=‖ f ‖q,α,2 .

(3) For all functions f ∈Lpq,α, where p ≥ 1 then Fq,αf ∈Lpq,α.
If 1 ≤ p ≤ 2 then

(10) ‖Fq,αf‖q,α,p ≤ B
2
p
−1

q,α ‖f‖q,α,p.
where

(11) Bq,α =
1

1 −q
(−q2; q2)∞(−q2α+2; q2)∞

(q2; q2)∞

(4) Let f ∈Lpq,α and g ∈Lrq,α the f ∗q g ∈Lsq,α and

(12) Fq,α(f ∗q,α g)(x) = Fq,αf(x) ×Fq,αf(x), ∀x ∈ R+q .
where 1 ≤ p,r,s such that

1

p
+

1

r
− 1 =

1

s
.

Proposition 2.4. [2] For all x,y ∈ R+q , we have

(13) Tαq,xjα(λy,q
2) = jα(λx,q

2)jα(λy,q
2).

Proposition 2.5. [2] For any function f ∈L2q,α we have

(14) Fq,α(Tαq,xf)(λ) = jα(λx,q
2)Fq,α(f)(λ), ∀λ,x ∈ R+q .



20 ABOUELAZ, DAHER AND LOUALID

3. Generalized q-Bessel transform

Let

• M the map defined by

(15) Mf(x) = x2nf(x).

• Lpq,α,n the class of measurable functions f on R+q for which

‖f‖q,α,p,n = ‖M−1f‖q,α+2n,p < ∞.

∀x ∈ R+q , put

(16) Ψα,n(λx,q2) = x2njα+2n(λx,q2).

Proposition 3.1. (i): The map M is a topological isomorphism from Lpq,α onto Lpq,α,n
(ii): We have

(17) ∆q,α,n ◦M = M◦ ∆q,α+2n.

(iii): Ψα,n(λ.,q2) satisfies the differential equation

(18) ∆q,α,nΨα,n(λ.,q2) = −λ2Ψα,n(λ.,q2)

Proof. Assertion (i) is easily checked.
(ii) easy combination of (1), (2) and (16).
Using (4) and (18), we have

∆q,α,nΨα,n(λ.,q
2) = M◦ ∆q,α+2n ◦M−1Ψα,n(λ.,q2),

= M◦ ∆q,α+2njα+2n(λ.,q2),
= −λ2Mjα+2n(λ.,q2),
= −λ2Ψα,n(λ.,q2)

which prove (iii).

Definition 3.2. The generalized q-Bessel transform of a function f ∈L1q,α,nis defined by

(19) Fq,α,n(f)(x) = cq,α+2n
∫ ∞
0

f(t)Ψα,n(λt,q
2)t2α+1dqt

where cq,α+2n is given by (3).

Proposition 3.3. (i): For all f ∈L1q,α,n we have

(20) Fq,α,n(f)(λ) = Fq,α+2n ◦M−1f(λ).

(ii): For all f ∈L1q,α,n

(21) Fq,α,n(∆q,α,nf)(λ) = −λ2Fq,α,n(f)(λ).

Proof. Let f ∈L1q,α,n. From (6), (17) and (20), we have

Fq,α,n(f)(λ) = cq,α+2n
∫ ∞
0

f(t)Ψα,n(λt,q
2)t2α+1dqt,

= cq,α+2n

∫ ∞
0

f(t)x2njα+2n(λt,q
2)t2α+1dqt,

= cq,α+2n

∫ ∞
0

M−1f(t)jα+2n(λt,q2)t2α+4n+1dqt,

= Fq,α+2n ◦M−1f(λ).



GENERALIZED q-BESSEL OPERATOR 21

which prove (i).
Let f ∈L1q,α,n. From (5), (19) and (20), we have

Fq,α,n(∆q,α,nf)(λ) = cq,α+2n
∫ ∞
0

∆q,α,nf(t)Ψα,n(λt,q
2)t2α+1dqt,

= cq,α+2n

∫ ∞
0

f(t)∆q,α,nΨα,n(λt,q
2)t2α+1dqt,

= cq,α+2n

∫ ∞
0

(−λ2)f(t)Ψα,n(λt,q2)t2α+1dqt,

= −λ2Fq,α,n(f)(λ).

Theorem 3.4. (1) For f ∈Lpq,α,n, we have
(22) ‖Fq,α,nf‖q,α,n,∞ ≤ Bq,α+2n‖f‖q,α,n,1.

where Bq,α+2n is given by (11)
(2) Let f ∈L1q,α,n, then

(23) ‖Fq,α,nf‖q,α,n,2 = ‖f‖q,α,n,2.

Proof. Let f ∈L1q,α,n, from (10), (11) and (21) we have

‖Fq,α,nf‖q,α,n,∞ = ‖Fq,α+2n ◦M−1f‖q,α+2n,∞,
≤ Bq,α+2n‖M−1f‖q,α+2n,1,
≤ Bq,α+2n‖f‖q,α,n,1.

which prove 1).
Let f ∈L1q,α,n. Using (9) and (21), we have

‖Fq,α,nf‖q,α,n,2 = ‖Fq,α+2n ◦M−1f‖q,α+2n,2,
= ‖M−1f‖q,α+2n,2,
= ‖f‖q,α,n,2.

4. Generalized convolution product associated with ∆q,α,n

Definition 4.1. The generalized q-Bessel translation operators Tαq,x,n associated with ∆q,α,n are
defined by

(24) Tαq,x,n = x
2nM◦Tα+2nq,x ◦M

−1

where Tα+2nq,x is given by (7).

The generalized q-Bessel translation operator is positive if

Tαq,x,nf ≥ 0, ∀f ≥ 0, ∀x ∈ R
+
q .

The domaine of positivity of the generalized q-Bessel translation operator is

Qα,n = {q ∈]0, 1[, Tαq,x,n is positive for all x ∈ R
+
q }.

In the rest of this paper we always assume that the generalized q-Bessel translation operator is positive.

Proposition 4.2. (i): Let f ∈L1q,α,n, we have
(25) Tαq,x,nf(y) = T

α
q,y,nf(x)

and
Tαq,x,nf(0) = f(x).

(ii): ∀x,y ∈ R+q , we have

(26) Tαq,x,nΨα,n(λy,q
2) = Ψα,n(λx,q

2)Ψα,n(λy,q
2)



22 ABOUELAZ, DAHER AND LOUALID

(iii): For any function f ∈L2q,α,n, we have

(27) Fq,α,n(Tαq,x,nf)(λ) = Ψα,n(λx,q
2)Fq,α,n(f)(λ).

Proof. Let f ∈L1q,α,n, from Porosition 2.2 and (25), we have

Tαq,x,nf(y) = x
2nM◦Tα+2nq,x ◦M

−1f(y),

= x2ny2nTα+2nq,y ◦M
−1f(x),

= y2nM◦Tα+2nq,y ◦M
−1f(x),

= Tαq,y,nf(x).

which prove (i).
Let x,y ∈ R+q . From (13) and (25), we have

Tαq,x,nΨα,n(λy,q
2) = x2nM◦Tα+2nq,x ◦M

−1(y2njα+2n(λy,q
2)),

= x2ny2nTα+2nq,x jα+2n(λy,q
2)),

= x2njα+2n(λx,q
2))y2njα+2n(λy,q

2)),

= Ψα,n(λx,q
2)Ψα,n(λy,q

2).

which prove (ii).
Let f ∈L2q,α,n. From (14), (17), (21) and (25), we have

Fq,α,n(Tαq,x,nf)(λ) = Fq,α,n(x
2nM◦Tα+2nq,x ◦M

−1f)(λ),

= x2nFq,α+2n(Tα+2nq,x ◦M
−1f)(λ),

= x2njα+2n(λx,q
2)Fq,α+2n(M−1f)(λ),

= Ψα,n(λx,q
2)Fq,α,n(f)(λ).

which prove (iii).

Definition 4.3. The generalized q-convolution product of both function f,g ∈L1q,α,n is defined by

(28) f ∗q,α,n g(x) = cq,α+2n
∫ ∞
0

Tαq,x,nf(y)g(y)y
2α+1dqy.

where cq,α+2n is given by (3).

Proposition 4.4. For f,g ∈L1q,α,n

(29) f ∗q,α,n g = M
[
(M−1f) ∗q,α+2n (M−1f)

]
.

Proof. Let f,g ∈L1q,α,n. From (8), (25) and (29), we have

f ∗q,α,n g(x) = cq,α+2n
∫ ∞
0

Tαq,x,nf(y)g(y)y
2α+1dqy,

= cq,α+2nx
2n

∫ ∞
0

Tα+2nq,x M
−1f(y)g(y)y2α+2n+1dqy,

= cq,α+2nx
2n

∫ ∞
0

Tα+2nq,x M
−1f(y)M−1g(y)y2α+4n+1dqy,

= x2n
[
(M−1f) ∗q,α+2n (M−1g)

]
(x),

= M
[
(M−1f) ∗q,α+2n (M−1g)

]
(x).

Proposition 4.5. For f,g ∈L1q,α,n, then f ∗q,α,n g ∈L1q,α,n and

(30) Fq,α,n(f ∗q,n g)(λ) = Fq,α,n(f)(λ)Fq,α,n(g)(λ).



GENERALIZED q-BESSEL OPERATOR 23

Proof. Let f,g ∈L1q,α,n, we have

‖f ∗q,n g‖q,α,n,1 = ‖M−1(f ∗q,n g)‖q,α+2n,1,
≤ ‖M−1f‖q,α+2n,1‖M−1g‖q,α+2n,1,
= ‖f‖q,α,n,1‖g‖q,α,n,1.

On the other hand, from (12), (21) and (30), we have

Fq,α,n(f ∗q,α,n g)(λ) = Fq,α,n
(
M
[
(M−1f) ∗q,α+2n (M−1f)

])
(λ),

= Fq,α+2n ◦M−1
(
M
[
(M−1f) ∗q,α+2n (M−1f)

])
(λ),

= Fq,α+2n
(
M−1f

)
(λ) ×Fq,α+2n

(
M−1g

)
(λ),

= Fq,α,n(f)(λ)Fq,α,n(g)(λ).

Proposition 4.6. Let f ∈L1q,α,n, we have

(31) Tαq,x,nf(y) =
∫ ∞
0

f(z)Dα,n(x,y,z)z
2α+1dqz,

where Dα,n(x,y,z) = c2q,α+2n
∫∞
0

Ψα,n(xt,q
2)Ψα,n(yt,q

2)Ψα,n(zt,q
2)t2α+4n+1dqt

Proof. Let f ∈L1q,α,n, from (7), (19) and (24) we have

Tαq,x,nf(y) = x
2n
(
M◦Tα+2nq,x ◦M

−1) (f)(y),
= x2ny2nTα+2nq,x ◦M

−1f(y),

= x2ny2ncq,α+2n

∫ ∞
0

Fq,α+2n ◦M−1f(t)jα+2n(xt,q2)jα+2n(yt,q2)t2α+4n+1dqt,

= cq,α+2n

∫ ∞
0

Fq,α,n(f)(t)Ψα,n(xt,q2)Ψα,n(yt,q2)t2α+4n+1dqt,

=

∫ ∞
0

f(z)

[
c2q,α+2n

∫ ∞
0

Ψα,n(xt,q
2)Ψα,n(yt,q

2)Ψα,n(zt,q
2)t2α+4n+1dqt

]
z2α+1dqz,

=

∫ ∞
0

f(z)Dα,n(x,y,z)z
2α+1dqz.

References

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Constructive Approximation, 34(2011), 435-472.

[2] Ahmed Fitouhi and M. Moncef Hamza, The q-jαBessel Function, Journal of Approximation Theory 115(2002),
144-166.

[3] F.H. Jackson, On a q-Defnite Integrals, Quarterly Journal of Pure and Application Mathematics, 41(1910), 193-203.
[4] G.Gasper and M.Rahman, Basic hypergeometric series, Encycopedia of mathematics and its applications, Volume

35, Cambridge university press (1990).
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Applied Mathematics, 7(2006), Article ID 171.
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Department of Mathematics, Faculty of Sciences Aïn Chock,, University of Hassan II, Casablanca,
Morocco

∗Corresponding author: mehdi.loualid@gmail.com