R E S E A R C H
Open Access
Generalized
q
-Bessel function and its
properties
Mansour Mahmoud
**Correspondence:
[email protected] 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 generalizedq-Bessel function, which is a generalization of the knownq-Bessel functions of kinds 1, 2, 3, and the newq-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 andq-difference equation, which gives us the differential equation of each of the Bessel function and the modified Bessel function whenqtends to 1. Finally, the quantum algebraE2(q) and its representations presented an algebraic derivation for
the generating function of the generalizedq-Bessel function.
MSC: Primary 33D45; 81R50; 22E70
Keywords: q-Bessel functions; generating function;q-difference equation; quantum algebra; irreducible representation
1 Introduction
Theq-shifted factorials are defined by []
(a;q)= ,
(a;q)k= k–
i=o
–aqk,
(a, . . . ,ar;q)k= r
j=
(aj;q)k; k= , , , . . . ,
(a;q)∞= ∞
i=
–aqi, wherea,ai’s,q∈Rsuch that <q< .
The one-parameter family ofq-exponential functions
E(qα)(z) = ∞
n= qαn
(q;q)n
zn
withα∈Rhas been considered in []. Consequently, in the limit whenq→, we have
limq→E(qα)(( –q)z) =ez. Exton [] presented the followingq-exponential functions:
E(μ,z;q) = ∞
n= qμn(n–)
[n]q!
zn,
where [n]q! =(–(q;qq))nn. 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 replacezby–xqandμby a, we get the followingq-exponential function:
Eq(x,a) =
∞
n= qa(n
)
(q;q)n
xn, ()
which satisfies the functional relation []
Eq(x,a) –Eq(qx,a) =xEq
qax,a,
which can be rewritten by the formula
DqEq(x,a) =
–qEq
qax,a, ()
where the Jacksonq-difference operatorDqis defined by []
Dqf(x) =
f(x) –f(qx)
( –q)x ()
and satisfies the product rule
Dq
f(x)g(x)=f(qx)Dqg(x) +g(x)Dqf(x).
There are two important special cases of the functionEq(x,a)
eq(x) =
∞
n= xn (q,q)n
, |x|< ()
and
Eq(x) =
∞
n= q(n)xn
(q,q)n
. ()
Theq-Bessel functions of kinds , and are defined by []
Jn()(x;q) = (q
n+;q) ∞ (q;q)∞
x
n
ϕ
, qn+
q; –x
, ()
Jn()(x;q) = (q
n+;q) ∞ (q;q)∞
x
n
ϕ
– qn+
q; –q
n+x
, ()
Jn()(x;q) = (q
n+;q)∞ (q;q)∞
x
n
ϕ
qn+
q; –q
n+ x
whererϕsis the basic hypergeometric function []
rϕs
a, . . . ,ar
b, . . . ,bs
q;z=∞
k=
(a, . . . ,ar;q)k
(b, . . . ,bs;q)k
(–)kqk(k–)+s–r z
k
(q;q)k
. ()
The functionsJn(i)(x;q),i= , , areq-analogues of the Bessel function, and the function
Jn()(x;q) is aq-analogue of the modified Bessel function.
Rogov [, ] introduced generalized modifiedq-Bessel functions, similarly to the classi-cal case [], as
Ini(x;q) =(q
n+;q)∞ (q;q)∞
x
n
δϕ
, , . . . , qn+
q;q
(n+)(–δ) x
,
where
δ=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
fori= ,
fori= ,
fori= .
Recently, Mansour andet al.[] studied the followingq-Bessel function:
Jn()(x;q) =(x/)
n
(q;q)n
ϕ
– ,qn+
q;–q
(n+) x
,
which is aq-analogy of the modified Bessel function.
In this paper, we define the generalizedq-Bessel function and study some of its prop-erties. Also, in analogy with the ordinary Lie theory [, ], we derive algebraically the generating function of the generalizedq-Bessel function.
2 The generalizedq-Bessel function and its generating function Definition . The generalizedq-Bessel function is defined by
Jn(x,a;q) =
(x/)n
(q;q)n
∞
k=
(–)k(a+)qak(k+n) (qn+;q)
k
(x/)k
(q;q)k
, ()
which converges absolutely for allxwhena∈Z+and for|x|< ifa= .
As special cases ofJn(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
Proof
lim
q→Jn
( –q)x,a;q=lim
q→
( –q)n (q,q)n
x
n∞
k=
(–)k(a+)qak(k+n) ( –q)
k
(qn+;q)
k(q;q)k
x
k
=
()n
x
n∞
k=
(–)k(a+) (n+ )k()k
x
k
=
x
n∞
k=
(–)k(a+)
(n+k+ )(k+ )
x
k
.
Hence, we get
lim
q→Jn
( –q)x,a;q=Jn(x); a= , , , . . . ()
and
lim
q→Jn
( –q)x,a;q=In(x); a= , , , . . . , ()
whereJn(x) is the Bessel function andIn(x) is the modified Bessel function.
Lemma . The function Jn(x,a;q)satisfies
J–n(x,a;q) = (–)n(a+)Jn(x,a;q), n∈Z. ()
Proof Using the definition (), we get
J–n(x,a;q) =
∞
k=n
(–)k(a+)qak(k–n)(q–n+k+;q) ∞ (q;q)∞(q;q)k
x
k–n
.
Fors=k–n, we obtain
J–n(x,a;q) =
∞
s=
(–)(s+n)(a+)qas(s+n)(qs+;q)∞
(q;q)∞(q;q)s+n
x
s+n
,
and using the relations []
qs+;q
∞=
qn+s+;q ∞
qs+;q
n,
(q;q)s+n= (q;q)s
qs+;qn,
we obtain
J–n(x,a;q) = (–)n(a+)
∞
s=
(–)s(a+)qas(s+n)(qn+s+;q)∞ (q;q)∞(q;q)s
x
s+n
Lemma . The function Jn(x,a;q)satisfies the relation
Jn(–x,a;q) = (–)nJn(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
which is a generating function of theq-Bessel functionJn()(x;q) [],
which is a generating function of theq-Bessel functionJn()(x;q) [],
which is a generating function of theq-Bessel functionJn()(x;q) and
gn(x,t, ;q) =Eq
which is a generating function of theq-Bessel functionJn()(x;q) [].
3 Theq-difference equation of the functionJn(x,a;q)
Now the generating function method [] will be used to deduce theq-difference equation of the generalizedq-Bessel function. Using equation (), we have
Eq
By applying the operatorDq, we get
(–)a+qah
Using equation (), we obtain
(–)a+qa+a(n+
Similarly, we can prove the following relation:
By using equations () and (), we obtain
(–)a+qn(a+)+ Jn+(√qxh,a;q) +Jn–(xh,a;q)
= (–)(a+)qan Jn+(xh,a;q) +q–nJn–(√qxh,a;q). ()
But
Jn–(√qxh,a;q) = (
√
qxh
)n– (q;q)∞
∞
k=
(–)k(a+)qak(k+n–)(qn+k;q)∞
(q;q)k
√
qxh
k
=q
n– (xh
)
n– (q;q)∞
∞
k=
(–)k(a+)qak(k+n–)(qn+k;q)
∞ (q;q)k
qk– + xh
k
=qn– J
n–(xh,a;q) – qn– (xh
)
n– (q;q)∞
∞
k=
(–)(k+)(a+)qa(k+)(k+n) (q;q)k
×qn+k+;q∞
xh
k+
=qn–
Jn–(xh,a;q) +
(–)aqan xh
Jn
qaxh,a;q.
Then
q–nJ
n–(√qxh,a;q) –Jn–(xh,a;q) =
(–)aqan xh
Jn
qaxh,a;q. ()
Equations () and () give the relation
q+nJn+(√qxh,a;q) –Jn+(xh,a;q) =–xh
Jn
qaxh,a;q. ()
Equations () and () give the relation
Dq+
qa(–n)
( –q)x
qnδ
q–
Jn(xh,a;q) =
(–)a+hq(a+)(n+)
( –q) Jn+
qa+ xh,a;q, ()
where the operatorδqis given byδqf(x) =f(√qx).
Now consider the following operator:
Mn,q=
Dq+
qa(–n)
( –q)x
qnδq– , ()
then we can rewrite equation () by the formula
Mn,qJn(xh,a;q) =
(–)a+hq(a+)(n+)
( –q) Jn+
qa+ xh,a;q. ()
Also, equations () and () give the relation
Dq+
q–a(+n)
( –q)x
q–nδ
q–
Jn(xh,a;q) =
hq(a+)(– n)
( –q) Jn–
If we consider the operator
then we can rewrite equation () by the formula
Nn,qJn(xh,a;q) =
hq(a+)(– n)
( –q) Jn–
qa+ xh,a;q. ()
Hence, theq-difference equation of the functionJn(x,a;q) takes the formula
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) =
x
–qn+Jn+
q–ax,a;q+ (–)a+q a(n+)
J
n+(x,a;q). () Now, if we replace aby 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
rela-tion.
Then we have two cases of the recurrence relation.
Case(): The functionJn(x,a;q) has the recurrence relation
Jn(x,a;q) =
x
–qn+J
n+(x,a;q) –qn+Jn+(x,a;q); ∀a= , , , . . . , () which is the recurrence relation of each ofJn()(x;q) andJn()(x;q).
Case(): The functionJn(x,a;q) has the recurrence relation
Jn(x,a;q) =q
n
x
–qn+–qnx
Jn+(x,a;q)
+qn+/Jn+(x,a;q); ∀a= , , , . . . , ()
which is the recurrence relation of each ofJn()(x;q) andJn()(x;q).
5 The quantum algebra approach toJn(x,a;q)
The quantum algebraEq() is determined by generatorsH,E+andE–with the commuta-tion relacommuta-tions
[H,E+] =E+, [H,E–] = –E–, [E–,E+] = . ()
By considering the irreducible representations (ω) ofEq() characterized byω∈C, then
the spectrum of the operatorHwill be the set of integersZ, and the basis vectorsfm,m∈Z,
satisfy
E±fm=ωfm±, Hfm=mfm, E+E–fm=ωfm, ()
where C=E+E– is the Casimir operator which commutes with the generatorsH, E+ andE–. The following differential operators presented a simple realization of (ω)
H=zd
dz, E+=ωz, E–=
ω
z ()
In the ordinary Lie theory, matrix elementsTsmof the complex motion group in the
representation (ω) are typically defined by the expansions [–]
eαE+eβE–eτHf
If we replace the mapping exby the mappingEq(x,a/) from the Lie algebra to the Lie
group with puttingτ = 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=
The combination between the two cases gives us the following expression:
Tsm(α,β) =q
Lemma .
Eq
αωz,a
Eq
βω
z, a
= ∞
s=–∞ qas
(–)(a+)α
β s
Js
q–aω(–)a+αβ,a;qzs, ()
where(–)a+αβ> . As special cases:
Considering () witha= ,α= –β= ,z=tandω=x
, we obtain the relation (). Considering () witha= ,α= –β= ,z=√t
qandω=
√
qx
, we obtain the relation ().
Considering () witha= ,α=β= ,z=tandω= √
qx
, we obtain the relation (). Considering () witha= ,α=β= ,z=tandω=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 References
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doi:10.1186/1687-1847-2013-121
