On the Bessel operator $odot _{B}^{t}$ related to the Bessel-Helmholtz and Bessel Klein-Gordon operator


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 4, 2018, 922-928
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On the Bessel operator �tB related to the
Bessel-Helmholtz and Bessel Klein-Gordon operator

Sudprathai Bupasiri

Department of Mathematics, Sakon Nakhon Rajabhat University, Sakon Nakhon 47000,
Thailand

Abstract. In this paper, we study the Bessel operator �t
B , iterated t-times and denote by

�t
B =

((
Ba1

+ · · ·+Bap
+m2

)2 − (Bap+1
+ · · ·+Bap+q

)2)t
where p+ q = n,Bai

= ∂2

∂a2
i

+ 2vi
ai

∂
∂ai

, 2vi = 2αi + 1, αi > − 1
2 , ai > 0, t ∈ Z+ ∪ {0}, m ∈ R+ ∪ {0}

and p+ q = n is the dimension of R+
n = {a : a = (a1, . . . , an), a1 > 0, . . . , an > 0}.

2010 Mathematics Subject Classifications: 46F10

Key Words and Phrases: Bessel Helmholtz operator, Bessel Klein-Gordon operator, Bessel
diamond operator

1. Introduction

Yildirim, Sarikaya and Ozturk [7] have showed that (−1)tS2t(a)∗R2t(a) is the solution
of the ♦tB

(
(−1)tS2t(a) ∗R2t(a)

)
= δ , where

♦tB =

( p∑
i=1

Bai

)2

−

 p+q∑
j=p+1

Baj

2t

. (1)

Here p + q = n,Bai = ∂2

∂a2i
+ 2vi

ai
∂
∂ai
, 2vi = 2αi + 1, αi > −1

2 , ai > 0, i = 1, 2, . . . , n,

t ∈ Z+ ∪{0} and n is the dimension of the R+
n = {a : a = (a1, . . . , an), a1 > 0, . . . , an > 0}

. Otherwise, the operator ♦kB can also be expressed in the form ♦tB = �t
B4t

B = 4t
B�

t
B,

where �t
B denote by

�t
B =

(
Ba1 +Ba2 + · · ·+Bap −Bap+1 −Bap+2 − · · · −Bap+q

)t
, (2)

DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3319

Email addresses: sudprathai@gmail.com (S. Bupasiri)

http://www.ejpam.com 922 c© 2018 EJPAM All rights reserved.



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (4) (2018), 922-928 923

and 4t
B denote by

4t
B = (Ba1 +Ba2 + · · ·+Ban)t . (3)

Now in this paper,

�tB =

 p∑
i=1

Bai −
p+q∑
j=p+1

Baj

+m2

t(
n∑
i=1

Bai +m2

)t
, p+ q = n. (4)

Thus
�tB =

(
�B +m2

)t (4B +m2
)t

=
(
4B +m2

)t (
�B +m2

)t
, (5)

where (
4B +m2

)t
=
(
Ba1 +Ba2 + · · ·+Ban +m2

)t
(6)

and (
�B +m2

)t
=
(
Ba1 +Ba2 + · · ·+Bap −Bap+1 − · · · −Bap+q +m2

)t
(7)

and from (4) with q = 0 and t = 1, we obtain

�B =
(
4B,p +m2

)2
,

where (
4B,p +m2

)
=
(
Ba1 +Ba2 + · · ·+Bap +m2

)
. (8)

Moreover for m = 0, then we obtain Bessel diamond operator and defined by (1).

2. Preliminaries

Denoted by T ba the generalized shift operator acting according to the law [2]

T baϕ(a) = C∗v

∫ π

0
. . .

∫ π

0
ϕ

(√
a2

1 + b21 − 2a1b1 cos θ1, . . . ,
√
a2
n + b2n − 2anbn cos θn

)
×
(
Πn
i=1 sin2vi−1

)
dθ1 . . . dθn,

where a, b ∈ R+
n , C

∗
v = Πn

i=1
Γ(vi+1)

Γ( 1
2)Γ(vi)

. We remark that this shift operator is closely con-

nected with the Bessel differential operator [2].

d2U

da2
+

2v

a

dU

da
=
d2U

db2
+

2v

b

dU

db

U(a, 0) = f(a),

Ub(a, 0) = 0.

The convolution operator determined by T ba is as follow:

(f ∗ ϕ) =

∫
R+
n

f(b)T baϕ(a)
(

Πn
i=1b

2vi
i

)
db. (9)

Convolution (9) is known as a B-convolution. We note the following properties for the
B-convolution and the generalized shift operator:



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (4) (2018), 922-928 924

(a) T ba · 1 = 1.

(b) T 0
a · f(a) = f(a).

(c) If f(a), g(a) ∈ C(R+
n ), g(a) is a bounded function, a > 0 and∫ ∞

0
|f(a)|

(
Πn
i=1a

2vi
i

)
da <∞,

then ∫
R+
n

T baf(a)g(b)
(

Πn
i=1b

2vi
i

)
db =

∫
R+
n

f(b)T bag(a)
(

Πn
i=1b

2vi
i

)
db.

(d) From (c), we have the following equality for g(a) = 1,∫
R+
n

T baf(a)
(

Πn
i=1b

2vi
i

)
db =

∫
R+
n

f(b)
(

Πn
i=1b

2vi
i

)
db

(e) (f ∗ g)(a) = (g ∗ f)(a).

Definition 1. ([6]) A distribution E is said to be a fundamental solution or an elementary
solution for the differential operator L if

LE = δ

, where δ is Dirac-delta distribution. Let L(D) be a differential operator with constant
coefficients. We say that a distribution E ∈ D′(Rn) is a fundamental solution or the
elementary solution of the differential operator L(D) if E satisfies L(D)E = δ in D′(Rn).

Lemma 1. If �t
Bu(a) = δ for a ∈ Γ+ = {a ∈ Rn : a1 > 0, a2 > 0, . . . , an > 0 and U > 0},

where �t
B is the Bessel ultra-hyperbolic operator iterated t-times defined by (2). Then

u(a) = R2t(a) is the unique elementary solution of the operator �t
B where

R2t(a) =
U (

2t−n−2|v|
2

)

yn(2t)
=

(∑p
i=1 a

2
i −

∑p+q
j=p+1 a

2
j

)( 2t−n−2|v|
2

)
yn(2t)

(10)

for

yn(2t) =
π

n+2|v|−1
2 Γ

(
2+2t−n−2|v|

2

)
Γ
(

1−2t
2

)
Γ(2t)

Γ
(

2+2t−p−2|v|
2

)
Γ(p−2t

2 )
, |v| =

n∑
i=1

vi. (11)

Lemma 2. Given the equation 4t
Bu(a) = δ for a ∈ R+

n , where 4t
B is the Laplace-Bessel

operator iterated t-times defined by (3). Then u(a) = (−1)tS2t(a) is an elementary solution
of the operator 4t

B where

S2t(a) =
|a|2t−n−2|v|

zn(2t)
(12)

for

zn(2t) =
Πn
i=12vi−

1
2 Γ
(
vi + 1

2

)
Γ(t)

2n+2|v|−4tΓ
(
n+2|v|−2t

2

) .



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (4) (2018), 922-928 925

Proof. The proofs of Lemma 1 and Lemma 2 are given in [7].

Lemma 3. Given the equation
(
�B +m2

)t
u(a) = δ for a ∈ R+

n , where
(
�B +m2

)t
is the

Bessel Klein-Gordon operator iterated t-times defined by equation (7), δ is the Dirac-delta
distribution, a ∈ R+

n and t ∈ Z+ ∪ {0}, then u(a) = FB,2t(a,m), where

FB,2t(a,m) =
∞∑
r=0

(
−t
r

)
m2rR2t+2r(a), (13)

R2t(a) is defined by (10).

Proof. See [5].

Lemma 4. Let �B be the Bessel ultra-hyperbolic operator, defined by (2) and δ is the
Dirac delta distribution for a ∈ R+

n , then(
�B +m2

)t
δ = FB,−2t(a,m),

where FB,−2t(a,m) is the inverse of FB,2t(a,m) in the convolution algebra.

Proof. Let
D(a) =

(
�B +m2

)t
δ,

convolving both sides by FB,2t(a,m), then

FB,2t(a,m) ∗D(a) = FB,2t(a,m) ∗
(
�B +m2

)t
δ

=
(
�B +m2

)t
FB,2t(a,m) ∗ δ

= δ. (14)

Since FB,2t(a,m) is lie in S′, where S′ is a space of tempered distribution, choose S′ ⊂ D′R,
where D′R is the right-side distribution which is a subspace of D′ of distribution. Thus
FB,2t(a,m) ∈ D′R, it follow that FB,2t(a,m) is an element of convolution algebra, thus by
([4], p.150-151), we have that the equation (14) has a unique solution

D(a) = FB,−2t(a,m) ∗ δ = FB,−2t(a,m). (15)

That complete the proof.

Lemma 5. Given the equation
(
4B +m2

)t
u(a) = δ for a ∈ R+

n , where
(
4B +m2

)t
is

the Bessel-Helmholtz operator iterated t-times defined by equation (6), δ is the Dirac-delta
distribution, a ∈ R+

n and t ∈ Z+ ∪ {0}, then u(a) = HB,2t(a,m) is an elementary solution

of the operator
(
4B +m2

)t
, where

HB,2t(a,m) =
∞∑
r=0

(
−t
r

)
m2r(−1)t+rS2t+2r(a), (16)

S2t(a) is defined by (12).



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (4) (2018), 922-928 926

Proof. See [9].

Lemma 6. The convolution FB,2t(a,m)∗HB,2t(a,m) exists and is a tempered distribution
where FB,2t(a,m) and HB,2t(a,m) be defined by (13) and (16), respectively.

Proof. From (13) and (16), we have

FB,2t(a,m) ∗HB,2t(a,m) =

( ∞∑
r=0

(
−t
r

)
m2rR2t+2r(a)

)

∗

( ∞∑
r=0

(
−t
r

)
m2r(−1)t+rS2t+2r(a)

)

=
∞∑
r=0

∞∑
s=0

(
−t
r

)(
−t
s

)
m2r+2s(−1)t+rS2t+2r(a) ∗R2t+2s(a).

Since the function S2t+2r(a) and R2t+2s(a) are tempered distributions, see( [3], p.302 and
[1], p.97). From ([10], p.152), the convolution of functions

(−1)t+rS2t+2r(a) ∗R2t+2s(a),

exists and is also a tempered distribution. Thus, FB,2t(a,m) ∗HB,2t(a,m) exists and also
is a tempered distribution.

3. Main results

Theorem 1. Given the equation

�tBT (a,m) = δ (17)

for a ∈ R+
n , where �tB is the Bessel operator iterated t-times defined by (5), then

T (a,m) = FB,2t(a,m) ∗HB,2t(a,m) (18)

is an elementary solution of (17), where FB,2t(a,m) and HB,2t(a,m) are defined by (13)
and (16), respectively, t ∈ Z+ ∪ {0} and m ∈ R+ ∪ {0}. Moreover, from (18) we obtain

FB,−2t(a,m) ∗ T (a,m) = HB,2t(a,m) (19)

as an elementary solution of the Bessel-Helmholtz operator (4B + m2)t iterated t-times
defined by (6) and in particular, for q = 0 then �tB reduces to the Bessel-Helmhotz operator(
4B,p +m2

)2t
of p-dimension iterated 2t-times and is defined by (8), where

4B,p = Ba1 +Ba2 + · · ·+Bap ,

thus (17) becomes (
4B,p +m2

)2t
T (a,m) = δ (20)



S. Bupasiri / Eur. J. Pure Appl. Math, 11 (4) (2018), 922-928 927

we obtain
T (a,m) = HB,4t(a,m) (21)

is an elementary solution of (20).

Proof. From (5) and (17) we have

�tBT (a,m) =
((

�B +m2
)t (4B +m2

)t)
T (a,m) = δ.

Convolution of the above equation by FB,2t(a,m) ∗ HB,2t(a,m) and the properties of
convolution with derivatives, we obtain(

�B +m2
)t
FB,2t(a,m) ∗

(
4B +m2

)t
HB,2t(a,m) ∗ T (a,m)

= FB,2t(a,m) ∗HB,2t(a,m) ∗ δ. (22)

Thus

T (a,m) = δ ∗ δ ∗ T (a,m) = FB,2t(a,m) ∗HB,2t(a,m) (23)

by Lemma 3 and Lemma 5. Now from (18) and by Lemma 3 and Lemma 4 and properties
of inverses in the convolution algebra, we obtain

FB,−2t(a,m) ∗ T (a,m) = δ ∗HB,2t(a,m) = HB,2t(a,m)

is an elementary solution of the Bessel-Helmhotz operator iterated t-times defined by (6).
In particular, for q = 0 then (17) becomes(

4B,p +m2
)2t

T (a,m) = δ (24)

where
(
4B,p +m2

)2t
is the Bessel-Helmholtz operator of p-dimension, iterated 2t-times

and is defined by (8). By Lemma 5, we have

T (a,m) = HB,4t(a,m) (25)

is an elementary solution of (17). This completes the proof.

Corollary 1. Given the equation

�tBT (a, 0) = δ (26)

for a ∈ R+
n , where �tB is the Bessel operator iterated t-times defined by (5), then

T (a, 0) = (−1)tS2t(a) ∗R2t(a) (27)

is an elementary solution of Bessel diamond operator, where R2t(a) and S2t(a) are defined
by (10) and (12), respectively.

Proof. If m = 0, then we have T (a, 0) = (−1)tS2t(a) ∗ R2t(a) yielding the result,, see
[7].



REFERENCES 928

Acknowledgements

The author would like to thank the referee for his suggestions which enhanced the pre-
sentation of the paper. The author was supported by Sakon Nakhon Rajabhat University
.

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