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Generalized q-Bessel function and its properties

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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 E 2 (q) 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]

( a ; q ) 0 = 1 , ( a ; q ) k = i = o k 1 ( 1 a q k ) , ( a 1 , , a r ; q ) k = j = 1 r ( a j ; q ) k ; k = 0 , 1 , 2 , , ( a ; q ) = i = 0 ( 1 a q i ) , where  a , a i ’s , q R  such that  0 < q < 1 .

The one-parameter family of q-exponential functions

E q ( α ) (z)= n = 0 q α n 2 2 ( q ; q ) n z n

with αR has been considered in [2]. Consequently, in the limit when q1, we have lim q 1 E q ( α ) ((1q)z)= e z . Exton [3] presented the following q-exponential functions:

E(μ,z;q)= n = 0 q μ n ( n 1 ) [ n ] q ! z n ,

where [ n ] q != ( q ; q ) n ( 1 q ) n . The relation between these two notations is given by

E(λ,z;q)= E q ( 2 λ ) ( q λ ( 1 q ) z ) .

In Exton’s formula, if we replace z by x 1 q and μ by 2a, we get the following q-exponential function:

E q (x,a)= n = 0 q a ( n 2 ) ( q ; q ) n x n ,
(1)

which satisfies the functional relation [4]

E q (x,a) E q (qx,a)=x E q ( q a x , a ) ,

which can be rewritten by the formula

D q E q (x,a)= 1 1 q E q ( q a x , a ) ,
(2)

where the Jackson q-difference operator D q is defined by [5]

D q f(x)= f ( x ) f ( q x ) ( 1 q ) x
(3)

and satisfies the product rule

D q ( f ( x ) g ( x ) ) =f(qx) D q g(x)+g(x) D q f(x).

There are two important special cases of the function E q (x,a)

e q (x)= n = 0 x n ( q , q ) n ,|x|<1
(4)

and

E q (x)= n = 0 q ( n 2 ) x n ( q , q ) n .
(5)

The q-Bessel functions of kinds 1, 2 and 3 are defined by [6]

(6)
(7)
(8)

where φ s r is the basic hypergeometric function [1]

φ s r ( a 1 , , a r b 1 , , b s | q ; z ) = k = 0 ( a 1 , , a r ; q ) k ( b 1 , , b s ; q ) k ( ( 1 ) k q k 2 ( k 1 ) ) 1 + s r z k ( q ; q ) k .
(9)

The functions J n ( i ) (x;q), i=1,2, are q-analogues of the Bessel function, and the function J n ( 3 ) (x;q) 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

I n i (x;q)= ( q n + 1 ; q ) ( q ; q ) ( x 2 ) n δ φ 1 ( 0 , 0 , , 0 q n + 1 | q ; q ( n + 1 ) ( 2 δ ) 2 x 2 4 ) ,

where

δ={ 2 for  i = 1 , 0 for  i = 2 , 1 for  i = 3 .

Recently, Mansour and et al. [10] studied the following q-Bessel function:

J n ( 4 ) (x;q)= ( x / 2 ) n ( q ; q ) n 0 φ 2 ( 0 , q n + 1 | q ; q 3 ( n + 1 ) 2 x 2 4 ) ,

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

J n (x,a;q)= ( x / 2 ) n ( q ; q ) n k = 0 ( 1 ) k ( a + 1 ) q a k 2 ( k + n ) ( q n + 1 ; q ) k ( x 2 / 4 ) k ( q ; q ) k ,
(10)

which converges absolutely for all x when a Z + and for |x|<2 if a=0.

As special cases of J n (x,a;q), we get

J n ( 1 ) ( x ; q ) = J n ( x , 0 ; q ) , J n ( 2 ) ( x ; q ) = J n ( x , 2 ; q ) , J n ( 3 ) ( x ; q ) = J n ( x , 1 ; q ) , J n ( 4 ) ( x ; q ) = J n ( x , 3 ; q ) .

Lemma 2.2 The function J n (x,a;q) is a q-analogy of each of the Bessel function and the modified Bessel function.

Proof

lim q 1 J n ( ( 1 q ) x , a ; q ) = lim q 1 { ( 1 q ) n ( q , q ) n ( x 2 ) n k = 0 ( 1 ) k ( a + 1 ) q a k ( k + n ) 2 ( 1 q ) 2 k ( q n + 1 ; q ) k ( q ; q ) k ( x 2 ) 2 k } = 1 ( 1 ) n ( x 2 ) n k = 0 ( 1 ) k ( a + 1 ) ( n + 1 ) k ( 1 ) k ( x 2 ) 2 k = ( x 2 ) n k = 0 ( 1 ) k ( a + 1 ) Γ ( n + k + 1 ) Γ ( k + 1 ) ( x 2 ) 2 k .

Hence, we get

lim q 1 J n ( ( 1 q ) x , a ; q ) = J n (x);a=0,2,4,
(11)

and

lim q 1 J n ( ( 1 q ) x , a ; q ) = I n (x);a=1,3,5,,
(12)

where J n (x) is the Bessel function and I n (x) is the modified Bessel function. □

Lemma 2.3 The function J n (x,a;q) satisfies

J n (x,a;q)= ( 1 ) n ( a + 1 ) J n (x,a;q),nZ.
(13)

Proof Using the definition (10), we get

J n (x,a;q)= k = n ( 1 ) k ( a + 1 ) q a k ( k n ) 2 ( q n + k + 1 ; q ) ( q ; q ) ( q ; q ) k ( x 2 ) 2 k n .

For s=kn, we obtain

J n (x,a;q)= s = 0 ( 1 ) ( s + n ) ( a + 1 ) q a s ( s + n ) 2 ( q s + 1 ; q ) ( q ; q ) ( q ; q ) s + n ( x 2 ) 2 s + n ,

and using the relations [1]

( q s + 1 ; q ) = ( q n + s + 1 ; q ) ( q s + 1 ; q ) n , ( q ; q ) s + n = ( q ; q ) s ( q s + 1 ; q ) n ,

we obtain

J n ( x , a ; q ) = ( 1 ) n ( a + 1 ) s = 0 ( 1 ) s ( a + 1 ) q a s ( s + n ) 2 ( q n + s + 1 ; q ) ( q ; q ) ( q ; q ) s ( x 2 ) 2 s + n = ( 1 ) n ( a + 1 ) J n ( x , a ; q ) .

 □

Lemma 2.4 The function J n (x,a;q) satisfies the relation

J n (x,a;q)= ( 1 ) n J n (x,a;q),nZ,
(14)

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 J n (x,a;q).

Theorem 1 The generating function g(x,t,a;q) of the function J n (x,a;q) is given by

g(x,t,a;q)= E q ( q a 4 x t 2 , a 2 ) E q ( ( 1 ) a + 1 q a 4 x 2 t , a 2 ) = n = q a n 2 4 J n (x,a;q) t n .
(15)

Proof

Let

g(x,t,a;q)= E q ( q a 4 x t 2 , a 2 ) E q ( ( 1 ) a + 1 q a 4 x 2 t , a 2 ) ,

then

g(x,t,a;q)= r = 0 s = 0 ( 1 ) s ( a + 1 ) q a 2 [ ( r 2 ) + ( s 2 ) ] + a 4 ( s + r ) ( q ; q ) r ( q ; q ) s ( x 2 ) s + r t r s .

For s=rn, we get

g(x,t,a;q)= n = r = 0 ( 1 ) ( r n ) ( a + 1 ) q a 2 [ ( r 2 ) + ( r n 2 ) ] + a 4 ( 2 r n ) ( q ; q ) r ( q ; q ) r n ( x 2 ) 2 r n t n .

Hence, for n0, the coefficient of t n is given by

c n = ( 1 ) n ( a + 1 ) q a n 2 4 ( x / 2 ) n ( q ; q ) n r = 0 ( 1 ) r ( a + 1 ) q a r 2 ( r n ) ( q ; q ) r ( q n + 1 ; q ) r ( x 2 ) 2 r ,

where ( q , q ) r n = ( q ; q ) n ( q n + 1 ; q ) r for n0. Then

c n = ( 1 ) n ( a + 1 ) q a n 2 4 J n (x,a;q)= q a n 2 4 J n (x,a;q).

Similarly, for n0. □

As special cases of g n (x,t,a;q), we obtain

g n (x,t,0;q)= E q ( x t 2 , 0 ) E q ( x 2 t , 0 ) = e q ( x t 2 ) e q ( x 2 t ) ,
(16)

which is a generating function of the q-Bessel function J n ( 1 ) (x;q) [13],

g n ( x , t q , 2 ; q ) = E q ( x t 2 , 1 ) E q ( q x 2 t , 1 ) = E q ( x t 2 ) E q ( q x 2 t ) ,
(17)

which is a generating function of the q-Bessel function J n ( 2 ) (x;q) [13],

g n (x,t,1;q)= E q ( q 4 x t 2 , 1 2 ) E q ( q 4 x 2 t , 1 2 ) ,
(18)

which is a generating function of the q-Bessel function J n ( 3 ) (x;q) and

g n (x,t,3;q)= E q ( q 3 / 4 x t 2 , 3 2 ) E q ( q 3 / 4 x 2 t , 3 2 ) ,
(19)

which is a generating function of the q-Bessel function J n ( 4 ) (x;q) [10].

3 The q-difference equation of the function J n (x,a;q)

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

E q ( q a 4 x t h 2 , a 2 ) E q ( ( 1 ) a + 1 q a 4 x h 2 t , a 2 ) = n = q a n 2 4 J n (xh,a;q) t n ;hR{0}.
(20)

By applying the operator D q , we get

( 1 ) a + 1 q a 4 h 2 ( 1 q ) t E q ( q a + 4 4 x t h 2 , a 2 ) E q ( ( 1 ) a + 1 q 3 a 4 x h 2 t , a 2 ) + q a 4 t h 2 ( 1 q ) E q ( q 3 a 4 x t h 2 , a 2 ) E q ( ( 1 ) a + 1 q a 4 x h 2 t , a 2 ) = n = q a n 2 4 D q J n ( x h , a ; q ) t n .

Using equation (20), we obtain

( 1 ) a + 1 q a 4 + a 2 ( n + 1 2 ) + n + 1 2 2 ( 1 q ) J n + 1 ( q a + 2 4 x h , a ; q ) + q a 4 + a n ( n 1 ) 4 2 ( 1 q ) J n 1 ( q a 4 x h , a ; q ) = q a n 2 4 h D q J n ( x h , a ; q ) .

Hence

(21)

Similarly, we can prove the following relation:

(22)

By using equations (21) and (22), we obtain

(23)

But

J n 1 ( q x h , a ; q ) = ( q x h 2 ) n 1 ( q ; q ) k = 0 ( 1 ) k ( a + 1 ) q a k 2 ( k + n 1 ) ( q n + k ; q ) ( q ; q ) k ( q x h 2 ) 2 k = q n 1 2 ( x h 2 ) n 1 ( q ; q ) k = 0 ( 1 ) k ( a + 1 ) q a k 2 ( k + n 1 ) ( q n + k ; q ) ( q ; q ) k ( q k 1 + 1 ) ( x h 2 ) 2 k = q n 1 2 J n 1 ( x h , a ; q ) q n 1 2 ( x h 2 ) n 1 ( q ; q ) k = 0 ( 1 ) ( k + 1 ) ( a + 1 ) q a ( k + 1 ) ( k + n ) 2 ( q ; q ) k × ( q n + k + 1 ; q ) ( x h 2 ) 2 k + 2 = q n 1 2 { J n 1 ( x h , a ; q ) + ( 1 ) a q a n 2 x h 2 J n ( q a 4 x h , a ; q ) } .

Then

q 1 n 2 J n 1 ( q xh,a;q) J n 1 (xh,a;q)= ( 1 ) a q a n 2 x h 2 J n ( q a 4 x h , a ; q ) .
(24)

Equations (23) and (24) give the relation

q 1 + n 2 J n + 1 ( q xh,a;q) J n + 1 (xh,a;q)= x h 2 J n ( q a 4 x h , a ; q ) .
(25)

Equations (21) and (25) give the relation

{ D q + q a ( 1 n ) 4 ( 1 q ) x [ q n 2 δ q 1 ] } J n (xh,a;q)= ( 1 ) a + 1 h q ( a + 2 ) ( n + 1 ) 4 2 ( 1 q ) J n + 1 ( q a + 2 4 x h , a ; q ) ,
(26)

where the operator δ q is given by δ q f(x)=f( q x).

Now consider the following operator:

M n , q = { D q + q a ( 1 n ) 4 ( 1 q ) x [ q n 2 δ q 1 ] } ,
(27)

then we can rewrite equation (26) by the formula

M n , q J n (xh,a;q)= ( 1 ) a + 1 h q ( a + 2 ) ( n + 1 ) 4 2 ( 1 q ) J n + 1 ( q a + 2 4 x h , a ; q ) .
(28)

Also, equations (22) and (24) give the relation

{ D q + q a ( 1 + n ) 4 ( 1 q ) x [ q n 2 δ q 1 ] } J n (xh,a;q)= h q ( a + 2 ) ( 1 n ) 4 2 ( 1 q ) J n 1 ( q a + 2 4 x h , a ; q ) .
(29)

If we consider the operator

N n , q = { D q + q a ( 1 + n ) 4 ( 1 q ) x [ q n 2 δ q 1 ] } ,
(30)

then we can rewrite equation (29) by the formula

N n , q J n (xh,a;q)= h q ( a + 2 ) ( 1 n ) 4 2 ( 1 q ) J n 1 ( q a + 2 4 x h , a ; q ) .
(31)

Hence, the q-difference equation of the function J n (x,a;q) takes the formula

M n 1 , q N n , q J n (xh,a;q)= ( 1 ) a + 1 q a + 2 4 h 2 4 ( 1 q ) 2 J n ( q a + 2 2 x h , a ; q ) .
(32)

If we replace h by 1q and consider the limit as q tends to 1, then we obtain

( 1 2 d d x n 1 2 x ) ( 1 2 d d x + n 2 x ) y(x)= ( 1 ) a + 1 4 y(x),
(33)

or

x 2 y (x)+x y ( n 2 + ( 1 ) a + 1 x 2 ) y=0.
(34)

The differential equation (34) gives the Bessel function at a=0,2,4, and the modified Bessel function at a=1,3,5, , which proves again that J n (x,a;q) is a q-analogy of each of them.

4 The recurrence relations of the function J n (x,a;q)

Lemma 4.1

J n (x,a;q)= 2 x ( 1 q n + 1 ) J n + 1 ( q a / 4 x , a ; q ) + ( 1 ) a + 1 q ( a + 2 ) ( n + 1 ) 2 J n + 2 (x,a;q).
(35)

Proof

J n ( x , a ; q ) = ( x / 2 ) n ( q ; q ) n k = 0 ( 1 ) k ( a + 1 ) q a k 2 ( k + n ) ( q ; q ) k ( q n + 1 ; q ) k + 1 ( 1 q n + 1 + q n + 1 q n + k 1 ) ( x 2 4 ) k = 2 x ( 1 q n + 1 ) ( x / 2 ) n + 1 ( q ; q ) n + 1 k = 0 ( 1 ) k ( a + 1 ) q a k 2 ( k + n + 1 ) ( q ; q ) k ( q n + 2 ; q ) k ( ( q a 4 x ) 2 4 ) k + q n + 1 ( x / 2 ) n + 2 ( q ; q ) n + 2 k = 1 ( 1 ) k ( a + 1 ) q a k 2 ( k + n ) ( q ; q ) k 1 ( q n + 3 ; q ) k 1 ( x 2 4 ) k 1 = 2 x ( 1 q n + 1 ) J n + 1 ( q a / 4 x , a ; q ) + ( 1 ) a + 1 q ( a + 2 ) ( n + 1 ) 2 ( x / 2 ) n + 2 ( q ; q ) n + 2 k = 0 ( 1 ) k ( a + 1 ) q a k 2 ( k + n + 2 ) ( q ; q ) k ( q n + 3 ; q ) k ( x 2 4 ) k = 2 x ( 1 q n + 1 ) J n + 1 ( q a / 4 x , a ; q ) + ( 1 ) a + 1 q ( a + 2 ) ( n + 1 ) 2 J n + 2 ( x , a ; q ) .

 □

Similarly, if we write (1 q k + q k q n + k 1 ) instead of (1 q n + 1 + q n + 1 q n + k 1 ), we can prove the following lemma.

Lemma 4.2

J n (x,a;q)= 2 x ( 1 q n + 1 ) J n + 1 ( q 2 a 4 x , a ; q ) + ( 1 ) a + 1 q a ( n + 1 ) 2 J n + 2 (x,a;q).
(36)

Now, if we replace a by a+2 in the recurrence relation (36), we get the recurrence relation (35). Then we have the following lemma.

Lemma 4.3 The two functions J n (x,a;q) and J n (x,a+2;q) have the same recurrence relation.

Then we have two cases of the recurrence relation.

Case (1): The function J n (x,a;q) has the recurrence relation

J n (x,a;q)= 2 x ( 1 q n + 1 ) J n + 1 (x,a;q) q n + 1 J n + 2 (x,a;q);a=0,2,4,,
(37)

which is the recurrence relation of each of J n ( 1 ) (x;q) and J n ( 2 ) (x;q).

Case (2): The function J n (x,a;q) has the recurrence relation

J n ( x , a ; q ) = q n 4 [ 2 x ( 1 q n + 1 ) q n 2 x 2 ] J n + 1 ( x , a ; q ) + q n + 1 / 2 J n + 2 ( x , a ; q ) ; a = 1 , 3 , 5 , ,
(38)

which is the recurrence relation of each of J n ( 3 ) (x;q) and J n ( 4 ) (x;q).

5 The quantum algebra approach to J n (x,a;q)

The quantum algebra E q (2) is determined by generators H, E + and E with the commutation relations

[H, E + ]= E + ,[H, E ]= E ,[ E , E + ]=0.
(39)

By considering the irreducible representations (ω) of E q (2) characterized by ωC, then the spectrum of the operator H will be the set of integers , and the basis vectors f m , mZ, satisfy

E ± f m =ω f m ± 1 ,H f m =m f m , E + E f m = ω 2 f m ,
(40)

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 d z , E + =ωz, E = ω z
(41)

acting on the space of all linear combinations of the functions z m , z a complex variable, mZ, with basis vectors f m (z)= z m .

In the ordinary Lie theory, matrix elements T s m of the complex motion group in the representation (ω) are typically defined by the expansions [79]

e α E + e β E e τ H f m = s = T s m (α,β,τ) f s .
(42)

If we replace the mapping e x by the mapping E q (x,a/2) from the Lie algebra to the Lie group with putting τ=0 in equation (42), we can use the model (41) to find the following q-analog of matrix elements of (ω):

E q (α E + ,a/2) E q (β E ,a/2) f m = s = T s m (α,β) f s ,
(43)

and hence

E q ( α ω z , a 2 ) E q ( β ω z , a 2 ) z m = r , t = 0 q a 2 [ ( r 2 ) + ( t 2 ) ] ( q ; q ) r ( q ; q ) t ω r + t α r β t z m t + r .

Now replace s by mt+r to get

E q ( α ω z , a 2 ) E q ( β ω z , a 2 ) z m = s = r = 0 q a 2 [ ( r 2 ) + ( m + r s 2 ) ] ( q ; q ) r ( q ; q ) m + r s ω m + 2 r s α r β m + r s z s

and by equating the coefficient of z s for ms on both sides, we get

T s m ( α , β ) = q a 2 ( m s 2 ) ( q ; q ) m s ( ω β ) m s r = 0 q a 2 r ( r + m s 1 ) ( q ; q ) r ( q m s + 1 ; q ) r ( ω 2 α β ) r , m s = q a 4 ( m s ) 2 ( ( 1 ) ( a + 1 ) β α ) m s 2 ( 1 ) ( m s ) ( a + 1 ) J m s ( 2 q a 4 ω ( 1 ) a + 1 α β , a ; q ) , ( 1 ) a + 1 α β > 0 ; m s = q a 4 ( s m ) 2 ( ( 1 ) ( a + 1 ) α β ) s m 2 J s m ( 2 q a 4 ω ( 1 ) a + 1 α β , a ; q ) , ( 1 ) a + 1 α β > 0 ; m s ,

where J n (x,a;q)= ( 1 ) n ( a + 1 ) J n (x,a;q).

Similarly,

T s m ( α , β ) = q a 4 ( s m ) 2 ( ( 1 ) ( a + 1 ) α β ) s m 2 J s m ( 2 q a 4 ω ( 1 ) a + 1 α β , a ; q ) , ( 1 ) a + 1 α β > 0 ; s m .

The combination between the two cases gives us the following expression:

T s m ( α , β ) = q a 4 ( s m ) 2 ( ( 1 ) ( a + 1 ) α β ) s m 2 J s m ( 2 q a 4 ω ( 1 ) a + 1 α β , a ; q ) , ( 1 ) a + 1 α β > 0 ,
(44)

which is valid for all m,sZ. Then we get the following result.

Lemma 5.1

(45)

where ( 1 ) a + 1 αβ>0.

As special cases:

Considering (45) with a=0, α=β=1, z=t and ω= x 2 , we obtain the relation (16).

Considering (45) with a=2, α=β=1, z= t q and ω= q x 2 , we obtain the relation (17).

Considering (45) with a=1, α=β=1, z=t and ω= q 4 x 2 , we obtain the relation (18).

Considering (45) with a=3, α=β=1, z=t and ω= q 3 / 4 x 2 , we obtain the relation (19).

References

  1. 1.

    Gasper G, Rahman M: Basic Hypergeometric Series. 2nd edition. Cambridge University Press, Cambridge; 2004.

  2. 2.

    Floreanini R, LeTourneux J, Vinet L: More on the q -oscillator algebra and q -orthogonal polynomials. J. Phys. A, Math. Gen. 1995, 28: L287-L293. 10.1088/0305-4470/28/10/002

  3. 3.

    Exton H: q-Hypergeometric Functions and Applications. Ellis Horwood, Chichester; 1983.

  4. 4.

    Atakishiyev NM: On a one-parameter family of q -exponential functions. J. Phys. A, Math. Gen. 1996, 29: L223-L227. 10.1088/0305-4470/29/10/001

  5. 5.

    Jackson FH: On q -functions and certain difference operator. Trans. R. Soc. Edinb. 1908, 46: 253–281.

  6. 6.

    Jackson FH: The application of basic numbers to Bessel’s and Legendre’s functions. Proc. Lond. Math. Soc. 1903–1904, 2: 192–220.

  7. 7.

    Rogov, V-BK: q-Bessel Macdonald functions. arXiv:math.QA/0010170v1, 17 Oct. 2000

  8. 8.

    Rogov, V-BK: The integral representations of the q-Bessel Macdonald functions. arXiv:math.QA/0101259v1, 31 Jan. 2001

  9. 9.

    Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG 2. In Higher Transcendental Functions. McGraw-Hill, New York; 1953. (based, in part, on notes left by Harry Bateman)

  10. 10.

    Mansour M, Al-Shomarani MM: New q -analogy of modified Bessel function and the quantum algebra E q (2) . J. Comput. Anal. Appl. 2013, 15(4):655–664.

  11. 11.

    Miller W: Lie Theory and Special Functions. Academic Press, San Diego; 1968.

  12. 12.

    Kalnins EG, Mukherjee S, Miller W: Models of q -algebra representations: the group of plane motions. SIAM J. Math. Anal. 1994, 25: 513–527. 10.1137/S0036141092224613

  13. 13.

    Dattoli G, Torre A: q -Bessel functions: the point of view of the generating function method. Rend. Mat. Appl. (7) 1997, 17: 329–345.

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Keywords

  • q-Bessel functions
  • generating function
  • q-difference equation
  • quantum algebra
  • irreducible representation