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Generalized q-Bessel function and its properties
Advances in Difference Equations volume 2013, Article number: 121 (2013)
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 and its representations presented an algebraic derivation for the generating function of the generalized q-Bessel function.
MSC:33D45, 81R50, 22E70.
1 Introduction
The q-shifted factorials are defined by [1]
The one-parameter family of q-exponential functions
with has been considered in [2]. Consequently, in the limit when , we have . Exton [3] presented the following q-exponential functions:
where . The relation between these two notations is given by
In Exton’s formula, if we replace z by and μ by 2a, we get the following q-exponential function:
which satisfies the functional relation [4]
which can be rewritten by the formula
where the Jackson q-difference operator is defined by [5]
and satisfies the product rule
There are two important special cases of the function
and
The q-Bessel functions of kinds 1, 2 and 3 are defined by [6]
where is the basic hypergeometric function [1]
The functions , , are q-analogues of the Bessel function, and the function is a q-analogue of the modified Bessel function.
Rogov [7, 8] introduced generalized modified q-Bessel functions, similarly to the classical case [9], as
where
Recently, Mansour and et al. [10] studied the following q-Bessel function:
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 [11, 12], we derive algebraically the generating function of the generalized q-Bessel function.
2 The generalized q-Bessel function and its generating function
Definition 2.1 The generalized q-Bessel function is defined by
which converges absolutely for all x when and for if .
As special cases of , we get
Lemma 2.2 The function is a q-analogy of each of the Bessel function and the modified Bessel function.
Proof
Hence, we get
and
where is the Bessel function and is the modified Bessel function. □
Lemma 2.3 The function satisfies
Proof Using the definition (10), we get
For , we obtain
and using the relations [1]
we obtain
□
Lemma 2.4 The function satisfies the relation
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 .
Theorem 1 The generating function of the function is given by
Proof
Let
then
For , we get
Hence, for , the coefficient of is given by
where for . Then
Similarly, for . □
As special cases of , we obtain
which is a generating function of the q-Bessel function [13],
which is a generating function of the q-Bessel function [13],
which is a generating function of the q-Bessel function and
which is a generating function of the q-Bessel function [10].
3 The q-difference equation of the function
Now the generating function method [13] will be used to deduce the q-difference equation of the generalized q-Bessel function. Using equation (15), we have
By applying the operator , we get
Using equation (20), we obtain
Hence
Similarly, we can prove the following relation:
By using equations (21) and (22), we obtain
But
Then
Equations (23) and (24) give the relation
Equations (21) and (25) give the relation
where the operator is given by .
Now consider the following operator:
then we can rewrite equation (26) by the formula
Also, equations (22) and (24) give the relation
If we consider the operator
then we can rewrite equation (29) by the formula
Hence, the q-difference equation of the function takes the formula
If we replace h by and consider the limit as q tends to 1, then we obtain
or
The differential equation (34) gives the Bessel function at and the modified Bessel function at , which proves again that is a q-analogy of each of them.
4 The recurrence relations of the function
Lemma 4.1
Proof
□
Similarly, if we write instead of , we can prove the following lemma.
Lemma 4.2
Now, if we replace a by in the recurrence relation (36), we get the recurrence relation (35). Then we have the following lemma.
Lemma 4.3 The two functions and have the same recurrence relation.
Then we have two cases of the recurrence relation.
Case (1): The function has the recurrence relation
which is the recurrence relation of each of and .
Case (2): The function has the recurrence relation
which is the recurrence relation of each of and .
5 The quantum algebra approach to
The quantum algebra is determined by generators H, and with the commutation relations
By considering the irreducible representations of characterized by , then the spectrum of the operator H will be the set of integers ℤ, and the basis vectors , , satisfy
where is the Casimir operator which commutes with the generators H, and . The following differential operators presented a simple realization of
acting on the space of all linear combinations of the functions , z a complex variable, , with basis vectors .
In the ordinary Lie theory, matrix elements of the complex motion group in the representation are typically defined by the expansions [7–9]
If we replace the mapping by the mapping from the Lie algebra to the Lie group with putting in equation (42), we can use the model (41) to find the following q-analog of matrix elements of :
and hence
Now replace s by to get
and by equating the coefficient of for on both sides, we get
where .
Similarly,
The combination between the two cases gives us the following expression:
which is valid for all . Then we get the following result.
Lemma 5.1
where .
As special cases:
Considering (45) with , , and , we obtain the relation (16).
Considering (45) with , , and , we obtain the relation (17).
Considering (45) with , , and , we obtain the relation (18).
Considering (45) with , , and , we obtain the relation (19).
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Mahmoud, M. Generalized q-Bessel function and its properties. Adv Differ Equ 2013, 121 (2013). https://doi.org/10.1186/1687-1847-2013-121
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DOI: https://doi.org/10.1186/1687-1847-2013-121