International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 1254

ISSN 2229-5518

Advanced q-Bessel Function of Two

Variables

Dr. Abha Tenguria, Ritu Sharma

Abstract :- The main object of this paper to construct q-Bessel functions of two variables of first kind and found various results of J r,s (x, y ; q) like generating function, recurrence relation. Furthermore we use q-analogue to find some new significant results and generalizations have been discovered.

Keyword:- Bessel function of one variable, Bessel function of two variable, differential equation, Factorial Notation, generating function, recurrence relations, q-analogy.

——————————  ——————————

I. INTRODUCTION

(a; q)n = �
1 n = 0

n−1

The problem of Mathematical Physics leads us to

determine the solutions of Differential Equations which satisfy certain prescribed conditions. Many special
(2)
�{(1 − aq )}

J=O

∞

for nϵN
functions have been exposed to current generalizations to a
base of q, which are usually noted as q –special function. Basic analogise of Bessel functions have been introduced by Jackson [4] and Swarthow[9].
we know that [1]the ordinary cylindrical Bessel functions, define q-generalization of power series expansions. Three
different types of such q-expansion can be recognized, each
(a; q)∞ = �{(1 − aqJ)}

J=O

(a; q)0 = 1
The q-factorial [n]q ! being defined as, where n is integer
(3)
of them satisfy recurrence relations, second order q- differential equation and addition theorems, which reduce to those holding for the usual Bessel function in the limit q
→ 1.This paper presents new form of Bessel function with q
–analogues of one and two variables.
Where 0 < q < 1.
[n]q ! =

(q; q)n

q 2

II. PRELIMINARY NOTATIONS AND DEFINITIONS

The Bessel’s function of first kind of order r is defined by [7]
(4)

∞

m
Jr (x) = �
x 2m+r

� �
Before entering the specific topic of the paper, let us briefly
review the properties of q – Bessel functions. We discuss

m=0

! m Γr + m + 1 2
Definitions and Notations of q-analogy.
The q – shifted factorial Notation of Real and Complex

————————————————

• Dr. Abha Tenguria is Professor In Govt. Maharani Laxmi Bai Girls' P.G.

The generalized q-Bessel function of one variable is defined by [1]
(5)

∞

(Autonomous) College Bhopal (M.P.), India.

(−1)m
x 2m+r

• Jr (x; q) = � [m]

[ r + m ] ! �2

• Ritu Sharma is Research Scholar In Atal Bihari Vajpayee Hindi

Vishwavidyalaya Bhopal (M.P.) , India

number is given by [1].
When q → 1, (4)

m=0 q ! q

(1)
And Two variable Bessel’s functions are defined by [7] with
r and s are integer we have the representations

International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 1255

ISSN 2229-5518

(6)

x
Eq ��

1
�t − � +
yρ(x)
�w −

1
��

x r y ρ(x)
2� 2 � x2

2 t 2 w
Jr,s (x, y) =
0F1 �−; r + 1; − � 0F1 �−; s

Γr + 1 Γs + 1 4
y2 ρ2(x)
for t ≠ 0, w ≠ 0 and t,w ∈ ℂ .
or
Jr,s (x, y)

∞

+ 1; − �
4
(−1)m+n
x 2m+r

�

y ρ(x) 2n+s
� �

Proof: - we know that by [2] there are two important special

cases of the q- exponential function
(8)

∞

xr
Eq (x) = � [r] !

r=0 q

and
= �

m,n=0

�
! n ! m Γr + m + 1 Γs + n + 1 2 2
(9)
In this paper we have to prove that q-Bessel function of two
variable generating functions, recurrence relation and some

∞

Eq (x) = �

qr �2 xr
〈q; q〉
results of

r=0 r

|x| < 1 .
∞ (−1)m+n �x�2m+r �yρ(x)

2n+s

Consequently in the limitq → 1, we have limq→1�Eq (1 −

2 2 �
q)z� = ez
by [2]
Jr,s (x, y ; q) = �

[m + r] ! [m] ! [n + s] ! [n] !

m,n=0 q q q q

Using the formula (8), we get L.H.S.

x
When q → 1 Eq ��
2

1
�t − t � +

yρ(x)
2
�w −

1
w��
xt x
yρ(x)w
yρ(x)

III. GENERATING FUNCTION FOR Jr,s(x ,y ;q)

= Eq �� 2 �� Eq ��− 2t�� Eq ��
2 �� Eq ��−

2w ��

Definition:-For the advanced q- Bessel function of two

variable Jr,s (x , y ; q) , we defined by
∞ ∞ �xt�

x m ∞

∞ �yρ(x)w

yρ(x) n
= � � 2 �– 2t�

2 � �–
� �

2w �
(7)

r=0 m=0

[r]q! [m]q !

s=0 n=0

[s]q ! [n]q !
∞ (−1)m+n �x�2m+r �yρ(x)�
If we replace r by m+ r & s by n + s, then we get following

2 2
Jr,s (x , y ; q) = �
[m + r] ! [m] ! [n + s] ! [n] !
equation

m,n=0 q q q q

m+r

∞ ∞ (−1)m � �

� x � ∞

∞ (−1)n �

yρ(x)w n+s

�

yρ(x) n
� �

= � � 2 2t � �
[m + r]q! [m]q !

2 2w
[n + s]q ! [n]q !
Jr,s (x , y ; q)

r=−∞ m=0

s=−∞ n=0

�x�r �yρ(x)�

2 2 2

∞ ∞ m+n

x m+r+m

yρ(x) n+s+n

n+s−n

m+r−m

= 2 2
−x
oF �−; [m] !; � oF
−y ρ(x)
�−; [n] !; �
(−1)
�2�
� 2 �
(w)
(t)
[r]q ! [s]q ! 1

q 4 1 q 4

= � �

[m + r] ! [m] ! [n + s] ! [n] !
Now we will deduce the generating function of the

r,s=−∞ m,n=0

q q q

2n+s

advanced q-Bessel function of two variables Jr,s(x,y;q).
∞ ∞ (−1)m+n �x�2m+r �yρ(x)�

= � t r ws � 2 2
Theorem 3.1 :- Prove that Jr,s (x,y;q) is the coefficient of t rws
in the expansion of

(10)

r,s=−∞

m,n=0

[m + r]q ! [m]q ! [n + s]q ! [n]q !

International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 1256

ISSN 2229-5518

x
Eq ��

1
�t − � +

yρ(x)
�w −

∞

1
�� = � t rws Jr,s (x , y ; q)
(14)
2 t 2
w

r,s=−∞

Γ q(n + 1) = [n]q!
Which is a generating function of Jr,s (x , y ; q) .If using (9)
then we get result.

∞

= lim �

2m+r

(−1)m+n(1 − q)2m(1 − q)2n � �

�yρ(x)�

q→1 [r]q! [s]q!

m,n=0

(q; q)m (qr+1; q)m(qs+1 ; q)n (q; q)n

(11)
x 1 yρ(x) 1
Taking limit q → 1 and we obtain

2n+s

Eq ��
2
�t −

t � +

2 �w −

∞

w��

1 ∞ (−1)m+n �x�
= �

2m+r

yρ(x)
� 2 �

r2 +s2

= q 2 � Jr,s (x , y ∶ t , w ; q)

r,s=−∞

[r]![s]!

m,n=0

(1)m (r + 1)m (s + 1)n (1)n
Theorem3.2 The function Jr,s (x , y ; q) is q –analogy of each
of the Bessel function an

∞ (−1)m+n �x�2m+r �yρ(x)�

1 2 2
= �
limq→1 Jr,s �(1 − q)x , (1 − q)y ; q� = Jr,s (x, y).
[r]! [s]!

m,n=0

Γm + 1 Γn + 1 Γr + m + 1 Γs + n + 1

Proof: - Some Previous result of q-analogy by Exton [6]

∞

(−1)m+n
x 2m+r

�

y ρ(x) 2n+s
� �
presented the q-exponential function, they say (12)

[n] = (q; q)n

q ! (1 − q)n

[n + k] = (q; q)n+k

q ! (1 − q)n+k

Where 0 < q < 1.
= �

m,n=0

�
! n ! m Γr + m + 1 Γs + n + 1 2 2
Hence, we get
(15)
lim Jr,s �(1 − q)x , (1 − q)y ; q� = Jr,s (x, y)

q→1

Using the relation by Gasper [8]
(13)
(q; q)n+k = (q; q)k (qk+1 ; q)n

IV. RECURRENCE RELATION FOR J r ,s (x ,y

;q)

lim Jr,s�(1 − q)x , (1 − q)y ; q�

q→1

2m+r

yρ(x) 2n+s

Lemma 4.1:-If r ,s be integer then Jr,s (x , y ; q)
satisfies

∞

= lim �

q→1

m,n=0

(−1)m+n(1 − q)m+r(1 − q)n(1 − q)m (1 − q)n+s �x �

(q; q)m+r (q; q)m(q; q)n+s(q; q)n

� 2 �

(16)
J−r,s
(x , y ; q) = (−1)r Jr,s
(x , y ; q)
Using the relation (13), we obtain

Proof: Using q-Bessel function (7)

∞ (−1)m+n �x�2m+r �yρ(x)�

2m+r

yρ(x) 2n+s

2 2
Jr,s (x , y ; q) = �

∞

= lim �

(−1)m+n (1 − q)2m (1 − q)2n (1 − q)r (1 − q)s �x �

� 2 �

m,n=0

[m + r]q ! [m]q ! [n + s]q ! [n]q !

q→1

m,n=0

(q; q)m (q; q)r (qr+1; q)m(qs+1 ; q)n (q; q)s (q; q)n

Here Substitute r → -r and we get

2m+r

yρ(x) 2n+s

= lim

(1 − q)r (1 − q)s

�

(−1)m+n (1 − q)2m (1 − q)2n �x�

� 2 �

q→1

(q; q)r (q; q)s

m,n=0

(q; q)m (qr+1 ; q)m (qs+1; q)n (q; q)n

2m−r

∞ m+n

2

yρ(x)
2 �

2n+s

We know that by relation of q-gamma function with q-
factorial function
J−r,s (x , y ; q) = � [m − r] ! [m] ! [n + s] ! [n] !

m,n=0 q q q q

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 1257

ISSN 2229-5518

Replace m by r +k then we obtain
Then Result is Obtain

2n+s r s

∞ (−1)r+k+n �x�2k+r �yρ(x)

= � 2 2
J−r,−s (x , y ; q) = (−1) Jr,−s(x , y ; q) = (−1) Jr,−s (x , y ; q)
= (−1)r+s Jr,s (x , y ; q)

k,n=0

[k + r]q ! [k]q ! [n + s]q ! [n]q !
∞ (−1)k+n �x�2k+r �yρ(x)�
Lemma 4.4:- The function Jr,s (x , y ; q)satisfies the relation
(19)

= (−1)r � 2 2
Jr,s (−x , y ; q) = (−1)
Jr,s (x , y ; q)

k,n=0

[k + r]q! [k]q ! [n + s]q ! [n]q !
and hence it is even (or odd) function if the integer n is even
(or odd).
= (−1)rJr,s (x , y ; q)
Which is the recurrence relation then we have the following
lemma (16)

Proof: We have by (7)

∞

(−1)m+n �x�2m+r �yρ(x)

2n+s

2 2 �
J−r,s (x , y ; q) = (−1)r Jr,s (x , y ; q)
Jr,s (x, y ; q) = �

[m + r] ! [m] ! [n + s] ! [n] !
Lemma 4.2:-If r, s be integer then Jr,s (x , y ; q) satisfies
(17)

m,n=0 q q

for x=-x, then we get

q q

2n+s

∞ (−1)m+n �−x 2m+r �yρ(x)

Jr,−s (x , y ; q) = (−1)s Jr,s (x , y ; q)
J (−x, y ; q) = �

2 � 2 �

Proof: Using Definition of q –Bessel function (7) and

Substitute s→ -s and we get
Then

r,s

m,n=0

[m + r]q ! [m]q ! [n + s]q ! [n]q !

2m+r

∞ (−1)m+n � �

�yρ(x)

2n−s

∞ (−1)m+n �x�

2m+r

yρ(x)

�

2n+s

2 2 �

2m+r � 2 2

Jr,−s (x , y ; q) = �
[m − r] ! [m] ! [n − s] ! [n] !
Jr,s (x, y ; q) = (−1)
[m + r] ! [m] ! [n + s] ! [n] !

m,n=0 q q q q

Replace n by s+ t then we get
∞ (−1)m+s+t �x�2k+r �yρ(x)�
For all value of m is positive(−1)2m = 1

= � 2 2
Then

t,n=0

[k + r]q ! [k]q ! [t]q! [s + t]q !

∞ (−1)m+n �x�2m+r �yρ(x)�

∞ (−1)m+t �x�2k+r �yρ(x)

2s+t

2 2
Jr,s (x, y ; q) = (−1)r �

= (−1)s �

2 2 �

m,n=0

[m + r]q ! [m]q ! [n + s]q ! [n]q !

t,n=0

[k + r]q! [k]q ! [t]q! [s + t]q !
Hence the recurrence relation is
Jr,−s (x , y ; q) = (−1)s Jr,s (x , y ; q)
Which is the prove of recurrence relation (17).
Lemma 4.3:-If r ,s be integer then Jr,s (x , y ; q) satisfies
(18)
J−r,−s (x , y ; q) = (−1)rJr,−s(x , y ; q) = (−1)s Jr,−s (x , y ; q)
= (−1)r+s Jr,s (x , y ; q)

Proof: Use (7 ) Substitute m→ r+k & n→ s+t

(−x , y ; q) = (−1)rJr,s (x , y ; q)

r,s

Lemma 4.5:- The function Jr,s (x , y ; q)satisfies the relation
(20)
Jr,s (x , −y ; q) = (−1)s Jr,s (x , y ; q)
and hence it is even (or odd) function if the integer n is even
(or odd).
Now, if we substitute y by –y in the relation (7) , then we get the result.

International Journal of Scientific & Engineering Research, Volume 5, Issue 12, December-2014 1258

ISSN 2229-5518

Lemma 4.6:- The function Jr,s (x , y ; q)satisfies the relation
(21)
Jr,s (−x , −y ; q) = (−1)r+s Jr,s (x , y ; q)
And hence it is even (or odd) function if the integer n is
even (or odd).
Now, if we substitute x by –x (same as above) & y by –y in
the relation (7) , then we get the result.

REFERENCES

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[2] Mansour Mahmoud, ‘Generalized q-Bessel function and its properties’ Advances in Difference Equations 2013, 2013:121.

[3] Math J., The zeros of basic Bessel functions, the functions Jν+ax(x),

and associated orthogonal polynomials, Anal. Appl. 86 (1982), no. 1, 1–

19.

[4] Jackson F. H., on generalized functions of Legendre and Bessel, Trans. Roy. Soc. Edin. 41 (1904), 1–28.

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[6] Exton H. q-Hypergeometric Functions and Applications. Ellis

Horwood, Chichester (1983)

[7] Rabia Aktas, Abdullah Altin, Bayram Cekim : On a Two –Variable Analogue Of The Bessel Functions. J. of Inequalities And Special Functions Issn: 2217-4303, vol-3(2012)pages 13-23

[8] Gasper G., Rahman M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004).

[9] Swarthow R.F., An addition Theorem And Some Product Formulas For The Hahn-Exton q –Bessel Function , Can . J. Math 44(1992), PP.867-879.

[10] Anderson G.D, Vamanamurthy M.K., Vuorinen M.: Special functions of quasiconformal theory, Exposition Math. 7(1989) 97–136.

[11] Watson G.N., A Treatise on the Theory of Bessel Functions, 2nd

Edition, Cambridge University Press, 1944.

[12] Abramowitz M., Stegun (Eds.) I.A., Handbook of Mathematical

Functions, 10th Edition, Dover Publications, Inc., New York, 1972.

[13] Jackson F. H., On generalized functions of Legendre and Bessel. Transactions of the Royal Society of Edinburgh, 41:l-28(1904).