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Mahmoud Advances in Difference Equations 2013, 2013:121
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Generalized q-Bessel function and its
properties
Mansour Mahmoud*
*
Correspondence:
mansour@mans.edu.eg
Mathematics Department, Faculty
of Science, King Abdulaziz
University, P.O. Box 80203, Jeddah,
21589, Saudi Arabia
Permanent address: Department of
Mathematics, Faculty of Science,
Mansoura University, Mansoura,
35516, Egypt
Abstract
In this paper, the generalized q-Bessel function, which is a generalization of the
known q-Bessel functions of kinds 1, 2, 3, and the new q-analogy of the modified
Bessel function presented in (Mansour and Al-Shomarani in J. Comput. Anal. Appl.
15(4):655-664, 2013) is introduced. We deduced its generating function, recurrence
relations and q-difference equation, which gives us the differential equation of each
of the Bessel function and the modified Bessel function when q tends to 1. Finally, the
quantum algebra E2 (q) and its representations presented an algebraic derivation for
the generating function of the generalized q-Bessel function.
MSC: Primary 33D45; 81R50; 22E70
Keywords: q-Bessel functions; generating function; q-difference equation; quantum
algebra; irreducible representation
1 Introduction
The q-shifted factorials are defined by []
(a; q) = ,
(a; q)k =
k–
 – aqk ,
i=o
(a , . . . , ar ; q)k =
r
(aj ; q)k ;
k = , , , . . . ,
j=
(a; q)∞ =
∞
 – aqi ,
where a, ai ’s, q ∈ R such that  < q < .
i=
The one-parameter family of q-exponential functions

Eq(α) (z) =
αn
∞
q  n
z
(q; q)n
n=
with α ∈ R has been considered in []. Consequently, in the limit when q → , we have
limq→ Eq(α) (( – q)z) = ez . Exton [] presented the following q-exponential functions:
E(μ, z; q) =
∞
qμn(n–)
n=
[n]q !
zn ,
© 2013 Mahmoud; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any
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where [n]q ! =
(q;q)n
.
(–q)n
Page 2 of 11
The relation between these two notations is given by
E(λ, z; q) = Eq(λ) q–λ ( – q)z .
In Exton’s formula, if we replace z by
function:
x
–q
and μ by a, we get the following q-exponential
n
∞
q a(  ) n
x ,
(q; q)n
n=
Eq (x, a) =
()
which satisfies the functional relation []
Eq (x, a) – Eq (qx, a) = xEq qa x, a ,
which can be rewritten by the formula
Dq Eq (x, a) =

Eq qa x, a ,
–q
()
where the Jackson q-difference operator Dq is defined by []
Dq f (x) =
f (x) – f (qx)
( – q)x
()
and satisfies the product rule
Dq f (x)g(x) = f (qx)Dq g(x) + g(x)Dq f (x).
There are two important special cases of the function Eq (x, a)
eq (x) =
∞
n=
xn
,
(q, q)n
|x| < 
()
and
Eq (x) =
n
∞
q (  ) xn
n=
(q, q)n
.
()
The q-Bessel functions of kinds ,  and  are defined by []
(qn+ ; q)∞ x n
x
,  ,
q; –
 ϕ
(q; q)∞


qn+ qn+ x
(qn+ ; q)∞ x n
– ()
q; –
,
Jn (x; q) =
 ϕ
(q; q)∞


qn+ n n+ 
n+
 x
(q
x
;
q)
q

∞
q; –
,
Jn() (x; q) =
 ϕ
(q; q)∞


qn+ Jn() (x; q) =
()
()
()
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where r ϕs is the basic hypergeometric function []
r ϕs
∞
(a , . . . , ar ; q)k +s–r zk
k
a , . . . , ar (–)k q  (k–)
.
q; z =
(b , . . . , bs ; q)k
(q; q)k
b , . . . , bs
k=
()
The functions Jn(i) (x; q), i = , , are q-analogues of the Bessel function, and the function
Jn() (x; q) is a q-analogue of the modified Bessel function.
Rogov [, ] introduced generalized modified q-Bessel functions, similarly to the classical case [], as
Ini (x; q) =
(qn+ ; q)∞ x n
, , . . . , 
δ ϕ
(q; q)∞

qn+
(n+)(–δ)
q  x
q;
,

where
⎧
⎪
⎪
⎨ for i = ,
δ =  for i = ,
⎪
⎪
⎩
 for i = .
Recently, Mansour and et al. [] studied the following q-Bessel function:
(x/)n
Jn() (x; q) =
 ϕ
(q; q)n
(n+)
– –q  x
,
q;

, qn+ which is a q-analogy of the modified Bessel function.
In this paper, we define the generalized q-Bessel function and study some of its properties. Also, in analogy with the ordinary Lie theory [, ], we derive algebraically the
generating function of the generalized q-Bessel function.
2 The generalized q-Bessel function and its generating function
Definition . The generalized q-Bessel function is defined by
Jn (x, a; q) =
ak
∞
(x/)n (–)k(a+) q  (k+n) (x /)k
,
(q; q)n
(qn+ ; q)k
(q; q)k
()
k=
which converges absolutely for all x when a ∈ Z+ and for |x| <  if a = .
As special cases of Jn (x, a; q), we get
Jn() (x; q) = Jn (x, ; q),
Jn() (x; q) = Jn (x, ; q),
Jn() (x; q) = Jn (x, ; q),
Jn() (x; q) = Jn (x, ; q).
Lemma . The function Jn (x, a; q) is a q-analogy of each of the Bessel function and the
modified Bessel function.
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Proof
∞
k k
ak(k+n)
( – q)n x n x
(
–
q)
(–)k(a+) q 
lim Jn ( – q)x, a; q = lim
q→
q→
(q, q)n 
(qn+ ; q)k (q; q)k 
k=
∞
 x n (–)k(a+) x k
=
()n 
(n + )k ()k 
k=
k
n ∞
(–)k(a+)
x
x
=
.

(n + k + )(k + ) 
k=
Hence, we get
lim Jn ( – q)x, a; q = Jn (x);
a = , , , . . .
()
lim Jn ( – q)x, a; q = In (x);
a = , , , . . . ,
()
q→
and
q→
where Jn (x) is the Bessel function and In (x) is the modified Bessel function.
Lemma . The function Jn (x, a; q) satisfies
n ∈ Z.
J–n (x, a; q) = (–)n(a+) Jn (x, a; q),
()
Proof Using the definition (), we get
J–n (x, a; q) =
ak(k–n)
∞
(–)k(a+) q  (q–n+k+ ; q)∞ x k–n
(q; q)∞ (q; q)k
k=n

.
For s = k – n, we obtain
J–n (x, a; q) =
as(s+n)
∞
(–)(s+n)(a+) q  (qs+ ; q)∞ x s+n
(q; q)∞ (q; q)s+n
s=

,
and using the relations []
= qn+s+ ; q ∞ qs+ ; q n ,
(q; q)s+n = (q; q)s qs+ ; q n ,
qs+ ; q
∞
we obtain
J–n (x, a; q) = (–)
n(a+)
as(s+n)
∞
(–)s(a+) q  (qn+s+ ; q)∞ x s+n
s=
= (–)n(a+) Jn (x, a; q).
(q; q)∞ (q; q)s

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Lemma . The function Jn (x, a; q) satisfies the relation
Jn (–x, a; q) = (–)n Jn (x, a; q),
n ∈ Z,
()
and hence it is even (or odd) function if the integer n is even (or odd).
Now we will deduce the generating function of the generalized q-Bessel function
Jn (x, a; q).
Theorem  The generating function g(x, t, a; q) of the function Jn (x, a; q) is given by
g(x, t, a; q) = Eq
a
a
∞
q  xt a
(–)a+ q  x a
an
,
,
Eq
=
q  Jn (x, a; q)t n .
 
t

n=–∞
()
Proof Let
g(x, t, a; q) = Eq
a
a
q  xt a
(–)a+ q  x a
,
,
Eq
,
 
t

then
g(x, t, a; q) =
s
a r
a
∞ ∞
(–)s(a+) q  [()+()]+  (s+r) x s+r
r= s=
(q; q)r (q; q)s

t r–s .
For s = r – n, we get
r–n
a r
a
∞
∞ (–)(r–n)(a+) q  [()+(  )]+  (r–n) x r–n n
g(x, t, a; q) =
t .
(q; q)r (q; q)r–n

n=–∞ r=
Hence, for n ≤ , the coefficient of t n is given by
cn = (–)–n(a+) q
an

ar
∞
(x/)–n (–)r(a+) q  (r–n) x r
,
(q; q)–n r= (q; q)r (q–n+ ; q)r 
where (q, q)r–n = (q; q)–n (q–n+ ; q)r for n ≤ . Then
cn = (–)–n(a+) q
an

J–n (x, a; q) = q
an

Jn (x, a; q).
Similarly, for n ≥ .
As special cases of gn (x, t, a; q), we obtain
gn (x, t, ; q) = Eq
xt
–x
xt
–x
,  Eq
,  = eq
eq
,

t

t
()
which is a generating function of the q-Bessel function Jn() (x; q) [],
xt
–qx
xt
–qx
t
,  Eq
,  = Eq
Eq
,
gn x, √ , ; q = Eq
q

t

t
()
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which is a generating function of the q-Bessel function Jn() (x; q) [],
√
√
 qxt 
 qx 
,
,
Eq
,
gn (x, t, ; q) = Eq
 
t 
()
which is a generating function of the q-Bessel function Jn() (x; q) and
gn (x, t, ; q) = Eq
/
q/ xt 
q x 
,
,
Eq
,


t 
()
which is a generating function of the q-Bessel function Jn() (x; q) [].
3 The q-difference equation of the function Jn (x, a; q)
Now the generating function method [] will be used to deduce the q-difference equation
of the generalized q-Bessel function. Using equation (), we have
Eq
a
a
∞
q  xth a
(–)a+ q  xh a
an
q  Jn (xh, a; q)t n ;
,
,
Eq
=


t

n=–∞
h ∈ R – {}.
()
By applying the operator Dq , we get
a+
a
a
(–)a+ q  h
q  xth a
(–)a+ q  xh a
Eq
,
,
Eq
( – q)t


t

a
a
a
q  th
q  xth a
(–)a+ q  xh a
+
Eq
,
,
Eq
( – q)


t

=
∞
q
an

Dq Jn (xh, a; q)t n .
n=–∞
Using equation (), we obtain
a a n+
n+
a an(n–)
a+
a
q+ 
(–)a+ q  +  (  )+ 
Jn+ q  xh, a; q +
Jn– q  xh, a; q
( – q)
( – q)
=
q
an

h
Dq Jn (xh, a; q).
Hence
Dq Jn (xh, a; q)
a
=
a+
a
n(a+)+
q h –an
(–)a+ q  Jn+ q  xh, a; q + q  Jn– q  xh, a; q .
( – q)
()
Similarly, we can prove the following relation:
Dq Jn (xh, a; q)
a
=
a
a+
(–an–n+)
q h na
(–)a+ q  Jn+ q  xh, a; q + q 
Jn– q  xh, a; q .
( – q)
()
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By using equations () and (), we obtain
√
Jn+ ( qxh, a; q) + Jn– (xh, a; q)
an
–n
√
= (–)(a+) q  Jn+ (xh, a; q) + q  Jn– ( qxh, a; q).
(–)a+ q
n(a+)+

()
But
√
Jn– ( qxh, a; q) =
=
(
√
qxh n– ∞
) 
(q; q)∞
q
k=
ak
(–)k(a+) q  (k+n–) (qn+k ; q)∞
(q; q)k
√
qxh

k
ak
∞
xh k
( xh
)n– (–)k(a+) q  (k+n–) (qn+k ; q)∞ k

q –+
(q; q)∞
(q; q)k

n–

k=
=q
n–

Jn– (xh, a; q) –
q
a(k+)(k+n)
∞
( xh
)n– (–)(k+)(a+) q 

(q; q)∞
(q; q)k
n–

k=
k+
xh
× qn+k+ ; q ∞

an
(–)a q  xh a
n–
= q  Jn– (xh, a; q) +
Jn q  xh, a; q .

Then
an
q
–n

(–)a q  xh a
√
Jn q  xh, a; q .
Jn– ( qxh, a; q) – Jn– (xh, a; q) =

()
Equations () and () give the relation
q
+n

–xh a
√
Jn+ ( qxh, a; q) – Jn+ (xh, a; q) =
Jn q  xh, a; q .

()
Equations () and () give the relation
a(–n)
(a+)(n+)
a+
q  n
(–)a+ hq 

Dq +
q δq –  Jn (xh, a; q) =
Jn+ q  xh, a; q ,
( – q)x
( – q)
()
√
where the operator δq is given by δq f (x) = f ( qx).
Now consider the following operator:
a(–n)
q  n
q  δq –  ,
Mn,q = Dq +
( – q)x
()
then we can rewrite equation () by the formula
(a+)(n+)
Mn,q Jn (xh, a; q) =
(–)a+ hq 
( – q)
a+
Jn+ q  xh, a; q .
()
Also, equations () and () give the relation
–a(+n)
(a+)(–n)
a+
q  –n
hq 
q  δq –  Jn (xh, a; q) =
Jn– q  xh, a; q .
Dq +
( – q)x
( – q)
()
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If we consider the operator
–a(+n)
q  –n
Nn,q = Dq +
q  δq –  ,
( – q)x
()
then we can rewrite equation () by the formula
(a+)(–n)
a+
hq 
Jn– q  xh, a; q .
Nn,q Jn (xh, a; q) =
( – q)
()
Hence, the q-difference equation of the function Jn (x, a; q) takes the formula
a+
Mn–,q Nn,q Jn (xh, a; q) =
(–)a+ q  h a+
Jn q  xh, a; q .
( – q)
()
If we replace h by  – q and consider the limit as q tends to , then we obtain
n–
 d
–
 dx
x
n
(–)a+
 d
+
y(x),
y(x) =
 dx x

()
or
x y (x) + xy – n + (–)a+ x y = .
()
The differential equation () gives the Bessel function at a = , , , . . . and the modified
Bessel function at a = , , , . . . , which proves again that Jn (x, a; q) is a q-analogy of each
of them.
4 The recurrence relations of the function Jn (x, a; q)
Lemma .
Jn (x, a; q) =
(a+)(n+)

 – qn+ Jn+ q–a/ x, a; q + (–)a+ q  Jn+ (x, a; q).
x
()
Proof
ak
∞
x k
(x/)n (–)k(a+) q  (k+n) n+
n+
n+k–
Jn (x, a; q) =
–q +q –q
(q; q)n
(q; q)k (qn+ ; q)k+

k=
=
ak
a
∞
(x/)n+ (–)k(a+) q  (k+n+) (q–  x) k

 – qn+
x
(q; q)n+
(q; q)k (qn+ ; q)k

k=
+ qn+
=
∞
n+ (x/)
(q; q)n+
k=
ak
(–)k(a+) q  (k+n) x k–
(q; q)k– (qn+ ; q)k– 

 – qn+ Jn+ q–a/ x, a; q
x
+ (–)
a+ (a+)(n+)

q
ak
∞
(x/)n+ (–)k(a+) q  (k+n+) x k
(q; q)n+
(q; q)k (qn+ ; q)k

k=
(a+)(n+)

=  – qn+ Jn+ q–a/ x, a; q + (–)a+ q  Jn+ (x, a; q).
x
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Similarly, if we write ( – qk + qk – qn+k– ) instead of ( – qn+ + qn+ – qn+k– ), we can prove
the following lemma.
Lemma .
Jn (x, a; q) =
–a
a(n+)

 – qn+ Jn+ q  x, a; q + (–)a+ q  Jn+ (x, a; q).
x
()
Now, if we replace a by a +  in the recurrence relation (), we get the recurrence
relation (). Then we have the following lemma.
Lemma . The two functions Jn (x, a; q) and Jn (x, a + ; q) have the same recurrence relation.
Then we have two cases of the recurrence relation.
Case (): The function Jn (x, a; q) has the recurrence relation
Jn (x, a; q) =

 – qn+ Jn+ (x, a; q) – qn+ Jn+ (x, a; q);
x
∀a = , , , . . . ,
()
which is the recurrence relation of each of Jn() (x; q) and Jn() (x; q).
Case (): The function Jn (x, a; q) has the recurrence relation
n
Jn (x, a; q) = q 

n x
 – qn+ – q  Jn+ (x, a; q)
x

+ qn+/ Jn+ (x, a; q);
∀a = , , , . . . ,
()
which is the recurrence relation of each of Jn() (x; q) and Jn() (x; q).
5 The quantum algebra approach to Jn (x, a; q)
The quantum algebra Eq () is determined by generators H, E+ and E– with the commutation relations
[H, E+ ] = E+ ,
[H, E– ] = –E– ,
[E– , E+ ] = .
()
By considering the irreducible representations (ω) of Eq () characterized by ω ∈ C, then
the spectrum of the operator H will be the set of integers Z, and the basis vectors fm , m ∈ Z,
satisfy
E± fm = ωfm± ,
E+ E– fm = ω fm ,
Hfm = mfm ,
()
where C = E+ E– is the Casimir operator which commutes with the generators H, E+
and E– . The following differential operators presented a simple realization of (ω)
H =z
d
,
dz
E+ = ωz,
E– =
ω
z
()
acting on the space of all linear combinations of the functions zm , z a complex variable,
m ∈ Z, with basis vectors fm (z) = zm .
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In the ordinary Lie theory, matrix elements Tsm of the complex motion group in the
representation (ω) are typically defined by the expansions [–]
∞
eαE+ eβE– eτ H fm =
Tsm (α, β, τ )fs .
()
s=–∞
If we replace the mapping ex by the mapping Eq (x, a/) from the Lie algebra to the Lie
group with putting τ =  in equation (), we can use the model () to find the following
q-analog of matrix elements of (ω):
Eq (αE+ , a/)Eq (βE– , a/)fm =
∞
Tsm (α, β)fs ,
()
s=–∞
and hence
t
a r
∞
a
ω a m q  [()+()] r+t r t m–t+r
Eq αωz,
Eq β ,
z =
ω αβz
.

z 
(q; q)r (q; q)t
r,t=
Now replace s by m – t + r to get
m+r–s
a r
∞ ∞
a
ω a m q  [()+(  )] m+r–s r m+r–s s
Eq β ,
z =
Eq αωz,
ω
αβ
z

z 
(q; q)r (q; q)m+r–s
s=–∞ r=
and by equating the coefficient of zs for m ≥ s on both sides, we get
Tsm (α, β) =
a m–s
a
∞
q(  )
q  r(r+m–s–)  r
ω αβ ,
(ωβ)m–s
(q; q)m–s
(q; q)r (qm–s+ ; q)r
r=
a
= q  (m–s)
m≥s
m–s
–a β 
(–)–(a+)
(–)(m–s)(a+) Jm–s q  ω (–)a+ αβ, a; q ,
α

(–)a+ αβ > ; m ≥ s
s–m

–a a (s–m)
(a+) α

(–)
Js–m q  ω (–)a+ αβ, a; q ,
=q
β
(–)a+ αβ > ; m ≥ s,
where J–n (x, a; q) = (–)n(a+) Jn (x, a; q).
Similarly,
Tsm (α, β) = q
a

 (s–m)
s–m

–a (a+) α
Js–m q  ω (–)a+ αβ, a; q ,
(–)
β
(–)a+ αβ > ; s ≥ m.
The combination between the two cases gives us the following expression:
Tsm (α, β) = q
(–)
a+
a

 (s–m)
(–)
(a+) α
β
s–m

–a Js–m q  ω (–)a+ αβ, a; q ,
αβ > ,
which is valid for all m, s ∈ Z. Then we get the following result.
()
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Lemma .
a
ω a
Eq β ,
Eq αωz,

z 
s
∞
a
α  –a q  s (–)(a+)
Js q  ω (–)a+ αβ, a; q zs ,
=
β
s=–∞
()
where (–)a+ αβ > .
As special cases:
Considering () with a = , α = –β = , z = t and ω = x ,√we obtain the relation ().
qx
Considering () with a = , α = –β = , z = √tq and ω =  , we obtain the relation ().
Considering () with a = , α = β = , z = t and ω =
Considering () with a = , α = β = , z = t and ω =
√
 qx
, we obtain the relation ().

q/ x
, we obtain the relation ().

Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Received: 23 February 2013 Accepted: 15 April 2013 Published: 29 April 2013
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doi:10.1186/1687-1847-2013-121
Cite this article as: Mahmoud: Generalized q-Bessel function and its properties. Advances in Difference Equations 2013
2013:121.