q-Laplace Type Transforms of q-Analogues of Bessel Functions

Journal of King Saud University – Science xxx (2018) xxx–xxx

Contents lists available at ScienceDirect

Journal of King Saud University – Science journal homepage: www.sciencedirect.com

Original article

q-Laplace Type Transforms of q-Analogues of Bessel Functions R.T. Al-Khairy Imam Abdulrahman Bin Faisal University, Eastern region, Al- Dammam, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 16 April 2018 Accepted 29 August 2018 Available online xxxx Keywords: q-extensions of Bessel functions q-Laplace Type Transforms q-shift factorials

a b s t r a c t In this work, q- Laplace type integral transforms which are called qL2 -transform and qL2 -transform are applied on three families of q2 -Bessel Functions. Moreover, we give some examples to show effectiveness of the proposed results. Ó 2018 Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction Bessel functions are series of solutions to a second order differential equation that arise in many diverse situations. In literature there are many q-extensions of Bessel functions. The first were introduced by Jackson (1905) and studied later by Hahn (1949), Exton and Srivastava (1994). These q-analogues have been studied extensively by Ismail in Ismail (1981,1982). Laplace transform is the most popular and widely used in applied mathematics. A certain type of Laplace transforms which is called L2 -transform was introduced by Yürekli and Sadek (1991). Then, these transforms were studied in more details by Yürekli (1999a,1999b). The qanalogue of L2 -transforms, which were called qL2 -transform and qL2 -transforms were studied by Uçar and Albayrak (2011) and applied to some basic functions. Purohit and Kalla applied the qLaplace transforms to a product of basic analogues of the Bessel functions Purohit and Kalla (2007). Uçar (2014) and Omari (2017) also applied Sumudu q-transforms and q-Natural transforms, respectively, to a product of q-analogues of the Bessel functions. In this paper, we evaluate qL2 -transform and qL2 -transforms presented in Uçar and Albayrak (2011) of a product of q2 analogues of three families of Bessel functions. Finally, some examples of different parameters are presented in order to illustrate the

Peer review under responsibility of King Saud University.

accuracy and potentialities of the given theorems. The obtained limiting case results show good agreement with the previously obtained solutions. 1.1. Definitions and preliminaries The mainly best known q-analogues of the remarkable Bessel function

J l ðxÞ ¼

1 X ð1Þk ðx=2Þlþ2k

k!Cðl þ k þ 1Þ

k¼0

:

ð1Þ

were given by Ismail (1982), ð1Þ

J l ðz; qÞ ¼

ð2Þ

J l ðz; qÞ ¼



1  z l X

2

z2 4

;

jzj < 2;

ð2Þ

;

z 2 C;

ð3Þ

n qz2 ; ðq; qÞlþn ðq; qÞn

z 2 C:

ð4Þ

n¼0

ðq; qÞlþn ðq; qÞn

1 qnðnþlÞ  z l X

ð3Þ

J l ðz; qÞ ¼ z

2 l

n¼0



z2 4

n

ðq; qÞlþn ðq; qÞn

1 X ð1Þn q n¼0

n

nðn1Þ 2



The q-shift factorials are defined, for fix a 2 C, as

ða; qÞ0 ¼ 1;

ða; qÞn ¼

n1 Y ð1  aqq Þ;

n ¼ 1; 2; . . . ;

ða; qÞ1

k¼0

¼ lim ða; qÞn : n!1

Production and hosting by Elsevier

We also denote by

E-mail address: [email protected] https://doi.org/10.1016/j.jksus.2018.08.012 1018-3647/Ó 2018 Production and hosting by Elsevier B.V. on behalf of King Saud University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article in press as: Al-Khairy, R.T. Journal of King Saud University – Science (2018), https://doi.org/10.1016/j.jksus.2018.08.012

ð5Þ

2

R.T. Al-Khairy / Journal of King Saud University – Science xxx (2018) xxx–xxx

9 x ½xq ¼ 1q ; x 2 C; > > 1q > = n n 2 N; ð½nq Þ! ¼ ððq;qÞ n ; 1qÞ > > > ða;qÞ ða; qÞx ¼ ðaq x ;a1Þ ; x 2 R: ;

ð1Þ

1

ð6Þ

ðq; qÞn

n¼0

¼ ðx; qÞ1 ;

constants then qL2 -transform of f ðtÞ is,

x 2 C:

ð7Þ

Whereas, the second type q-analogue of the exponential function was introduced as

eq ðxÞ ¼

1 X

xn 1 ¼ ; jxj < 1: ðx; qÞ1 ðq; qÞ n n¼0

ð8Þ

The series representations of the q-gamma function are given by Kac and De Sole (2005)

Cq ðaÞ ¼

1 ðq; qÞ1 X qka ; a1 ðq; qÞk ð1  qÞ k¼0

L2 ff ðtÞ; sg ¼

tet

2 s2

ð2Þ

where D; ak and lk where k ¼ 1; 2; . . . ; m are constants, then qL2 transform of f ðtÞ is, q L2 ff ðtÞ; sg ¼

¼

a 2 R:

f ðtÞdt:

ð11Þ

ð12Þ

1 1 ðq2 ; q2 Þ1 X q2n f ðqn s1 Þ: 2 ; q2 Þ ½ 2 s2 ðq n n¼0

ð13Þ

X 1 1 q2n ðs2 ; q2 Þn f ðqn Þ: 2 2 ½2 ðs ; q Þ1 n2Z

 1 L 2 f ðt 1=2 Þ; s2 : q L2 ff ðtÞ; sg ¼ ½ 2 q Also, similar relation exists between Lq -transform as follows:

¼

 1 L 2 f ðt 1=2 Þ; s2 : ½2 q

ð3Þ

q L2 ff ðtÞ; sg ¼

ð14Þ

ð3Þ

lk where k ¼ 1; 2; . . . ; m, then, qL2 -transform of

where D; ak and f ðtÞ is,

There exists a relation between q L2 -transform and Lq -transform as follows:

q L2 ff ðtÞ; sg

j m 1  lk Þ Y X a2 k  2 jk ðjk þ2 2 Bk ðs;2Þ  k2 q Hjk ðq2 ÞCq2 ðlk þ D þ jk Þ: 4s j ¼0 k¼1

ð10Þ

where, using (6), ½2 ¼ ½2q ¼ 1 þ q. While the series form of the second type of the q-analogue of L2 -transform which is denoted by q L2 as given by Uçar and Albayrak (2011) q L2 ff ðtÞ; sg

ð2Þ

(c) Let J 2l1 ða1 t; q2 Þ; J 2l2 ða2 t; q2 Þ; . . . ; J 2lm ðam t; q2 Þ be a set of q2 Q ð3Þ 2 Bessel functions of the third kind and f ðtÞ ¼ t2D2 m k¼1 J 2lk ðak t; q Þ

The series form of the first type of the q-analogue of L2 -Laplace transform which is denoted by q L2 as given by Uçar and Albayrak (2011)

¼

ð2Þ

(b) Let J 2l1 ða1 t; q2 Þ; J 2l2 ða2 t; q2 Þ; . . . ; J 2lm ðam t; q2 Þ be a set of q2 Q ð2Þ 2 Bessel functions of the second kind and f ðtÞ ¼ t2D2 m k¼1 J 2lk ðak t; q Þ

ð18Þ

0

q L2 ff ðtÞ; sg

ð17Þ

k

L2 -Laplace transform as defined by Yürekli and Sadek (1991) is written as 1

j m 1  Y X a2 k Bk ðs; 2Þ  k2 Hjk ðq2 ÞCq2 ðlk þ D þ jk Þ: 4s j ¼0 k¼1

ð9Þ

where KðA; aÞ is a remarkable function given by

Z

¼

ð3Þ

Xqk a  1  Cq ðaÞ ¼  ;q : A ð1  qÞa1 ð1 ; qÞ1 k2Z A k A ðq=a; qÞ1 ða; qÞ1 ; ðqt =a; qÞ1 ðaq1t ; qÞ1

q L2 ff ðtÞ; sg

k

KðA; aÞ

KðA; aÞ ¼ Aa1

m

of q -Bessel functions of the first kind and f ðtÞ ¼ Q ð1Þ 2 t2D2 m k¼1 J 2l ðak t; q Þ where D; ak and lk where k ¼ 1; 2; . . . ; m are set

The first type q-analogue of the exponential function was introduced as

Eq ðxÞ ¼

ð1Þ

2

2

k

1

nðn1Þ 1 X ð1Þn q 2 xn

ð1Þ

Theorem 1. (a) Let J 2l ða1 t; q2 Þ; J 2l ða2 t; q2 Þ; . . . ; J 2l ðam t; q2 Þ be a

j m 1  Y X a2 q2 k  2 jk ðjk21Þ Bk ðs;1Þ  k2 q Hjk ðq2 ÞCq2 ðlk þ D þ jk Þ: s j ¼0 k¼1 k

ð19Þ where ReðsÞ > 0, ReðDÞ > 0 and

Bk ðs; sÞ ¼

Hjk ðqÞ ¼

2lk

ak

½222lk ðs1Þ s2Dþ2lk

;

ð20Þ

ð1  qÞlk þDþjk 1 : ðq; qÞjk þ2lk ðq; qÞjk

ð21Þ

Proof. We will only give the proof of (17), because the proof of (18) and (19) is the same. We put

f ðtÞ ¼ t 2D2

m Y

ð1Þ

J 2lk ðak t; q2 Þ:

ð22Þ

k¼1

into the definition (13) yields

ð15Þ q L2 -transform

and

qL2 ff ðtÞ; sg ¼

m X 1 ðq2 ; q2 Þ1 Y q2n 2D2 ð1Þ ðqn s1 Þ J 2lk ðak ðqn s1 Þ; q2 Þ 2 ½2s ðq2 ; q2 Þn k¼1 n¼0

m X 1 ðq2 ; q2 Þ1 Y q2nD ð1Þ J ðak ðqn s1 Þ; q2 Þ 2 D ½2s ðq2 ; q2 Þn 2lk k¼1 n¼0 2 3  jk n 1 2  2lk X m X 1 1  ðak q 4s Þ ðq2 ; q2 Þ1 Y q2nD 6 ak ðqn s1 Þ 7 ¼ 4 5 2 ½2s2D k¼1 n¼0 ðq2 ; q2 Þn ðq2 ; q2 Þjk þ2lk ðq2 ; q2 Þjk j ¼0

¼

ð16Þ

k

1.2. Main Theorems In this section we evaluate qL2 -transform and qL2 -transform of t2D2 weighted product of m different q2 -Bessel Functions. The q2 Bessel Functions are more relevant than the original q-Bessel Functions because of the mathematical nature of qL2 -transform and qL2 -transform which contain q2 -shift factorials.

¼

ak 2lk n¼0 2nDþ2nl 1 Xq k X 2 2 D þ2 lk ½2s ðq2 ; q2 Þn j ¼0 ðq2 ; q2 Þjk þ2lk ðq2 ; q2 Þjk 1 k

m Y ðq2 ; q2 Þ1 k¼1

 jk n 1 2  ðak q 4s Þ

:

On interchanging the order of summations, which is valid under the conditions given by the theorem, we obtain

Please cite this article in press as: Al-Khairy, R.T. Journal of King Saud University – Science (2018), https://doi.org/10.1016/j.jksus.2018.08.012

3

R.T. Al-Khairy / Journal of King Saud University – Science xxx (2018) xxx–xxx

qL2 ff ðtÞ; sg ¼

m Y

Bk ðs; 2Þ

k¼1

1 X

 a2 jk  4sk2

jk ¼0

ðq2 ; q2 Þjk þ2lk ðq2 ; q2 Þjk

qL2 ff ðtÞ; sg ¼

1 X ðq2 ; q2 Þ1

m X Y 1 2D2 ð1Þ q2n ðs2 ; q2 Þn ðqn Þ J2lk ðak qn ; q2 Þ ½2ðs2 ; q2 Þ1 k¼1 n2z

m X Y 1 ð1Þ q2nD ðs2 ; q2 Þn J2lk ðak qn ; q2 Þ ½2ðs2 ; q2 Þ1 k¼1 n2z 2 3   n 2 jk  2lk X m X 1 n  ðak q4 Þ Y 1 7 2nD 2 2 6 ak ðq Þ ¼ q ðs ; q Þn 4 5 ðq2 ; q2 Þjk þ2lk ðq2 ; q2 Þjk ½2ðs2 ; q2 Þ1 k¼1 n2z 2 j ¼0

¼

n¼0

 2 nðDþlk þjk Þ q ðq2 ; q2 Þn  a2 jk Dþl þj 1 m 1  4sk2 ð1  q2 Þ k k Y X ¼ Bk ðs; 2Þ Cq 2 ð l k þ D ðq2 ; q2 Þjk þ2lk ðq2 ; q2 Þjk j ¼0 k¼1 

k

¼

k¼1

k

þ jk Þ

j 1  X a2 k ¼ Bk ðs; 2Þ  k2 Hjk ðq2 ÞCq2 ðlk þ D þ jk Þ:  4s j ¼0 k¼1 m Y

qL2 ff ðtÞ; sg ¼ ð1Þ

ð1Þ

qL2 ff ðtÞ; sg ¼

j 1  X H ðq2 Þ a2 k  1 jk  C 2 ðl þ D þ jk Þ: Bk ðs; 2Þ  k2 4s K ; lk þ D þ jk q k s2 j ¼0 k¼1

m Y

X q2nDþ2nlk ðs2 ; q2 Þn

  n 2 jk  ðak q4 Þ

n2z

jk ¼0

ðq2 ; q2 Þjk þ2lk ðq2 ; q2 Þjk

:

ak 2lk 2

½2

jk



jk

  X ðs2 ; q2 Þn q2 nðDþlk þjk Þ : ðq2 ; q2 Þjk þ2lk ðq2 ; q2 Þjk n2z ðs2 ; q2 Þ1 ¼0

1 X



a2k 4

Now, use Eq. (10) with A ¼ s12 and a ¼ D þ lk þ jk then, the summation on n can be written as

  X ðs2 ; q2 Þn q2 nðDþlk þjk Þ Cq2 ðlk þ D þ jk Þð1  q2 ÞDþlk þjk 1   ¼ ; ðs2 ; q2 Þ1 K s12 ; lk þ D þ jk s2ðlk þDþjk Þ n2z then

k

ð23Þ

m Y k¼1

ð1Þ

where D; ak and lk where k ¼ 1; 2; . . . ; m are constants then qL2 transform of f ðtÞ is,

k

2

½2ðs2 ; q2 Þ1

1 X

On interchanging the order of summations, which is valid under the conditions given by the theorem, we obtain

k

Theorem 2. (a) Let J 2l1 ða1 t; q2 Þ; J 2l2 ða2 t; q2 Þ; . . . ; J 2lm ðam t; q2 Þ be a set Q ð1Þ 2 of q2 -Bessel function of the first kind and f ðtÞ ¼ t 2D2 m k¼1 J 2lk ðak t; q Þ

a 2lk

m Y

qL2 ff ðtÞ; sg ¼

j m 1  Y X H ðq2 Þ a2 k  1 jk  C 2 ðl þ D þ jk Þ:  Bk ðs; 2Þ  k2 4s K ; l k þ D þ jk q k 2 s j ¼0 k¼1 k

ð2Þ

ð2Þ

ð2Þ

(b) Let J 2l1 ða1 t; q2 Þ; J 2l2 ða2 t; q2 Þ; . . . ; J 2lm ðam t; q2 Þ be a set of q2 Q ð2Þ 2 Bessel functions of the second kind and f ðtÞ ¼ t 2D2 m k¼1 J 2lk ðak t; q Þ where D; ak and lk where k ¼ 1; 2; . . . ; m are constants then qL2 transform of f ðtÞ is, lk Þ j  jk ðjk þ2 m 1  2 Y X Hjk ðq2 Þ a2k k q2 1  Cq2 ðlk þ D þjk Þ: qL2 ff ðtÞ;sg ¼ Bk ðs;2Þ  2 4s ; l þ D þjk K k s2 j ¼0 k¼1 k

ð24Þ ð3Þ

ð3Þ

ð3Þ

(c) Let J 2l1 ða1 t; q2 Þ; J 2l2 ða2 t; q2 Þ; . . . ; J 2lm ðam t; q2 Þ be a set of q2 -BesQ ð3Þ 2 sel function of the third kind and f ðtÞ ¼ t2D2 m k¼1 J 2lk ðak t; q Þ where

D; ak and lk where k ¼ 1; 2; . . . ; m, then, qL2 -transform of f ðtÞ is,

qL2 ff ðtÞ; sg ¼

j 1  X a2 q2 k Bk ðs; 1Þ  k2 s j ¼0 k¼1

In this final section we mainly give qL2 -transforms and qL2 transforms involving the q2 Bessel Functions as applications of our main results. Corollary 3. If one takes m ¼ 1; l1 ¼ l and a1 ¼ a in above theorems, respectively, one has

j 1  n o X a2 ð1Þ qL2 t2D2 J 2l ðat; q2 Þ; s ¼ Bðs; 2Þ  2 Hj ðq2 ÞCq2 ðl þ D þ jÞ; 4s j¼0 j 1  n o X a2  2 jðjþ22 lÞ ð2Þ qL2 t2D2 J 2l ðat;q2 Þ; s ¼ Bðs;2Þ  2 q Hj ðq2 ÞCq2 ðl þ D þ jÞ; 4s j¼0

m Y

j 1  n o X a2 q2  2 jðj1Þ ð3Þ  2 q 2 Hj ðq2 ÞCq2 ðl þ D þ jÞ; qL2 t2D2 J 2l ðat;q2 Þ; s ¼ Bðs;1Þ s j¼0

k

 2 jk ðjk21Þ q Hjk ðq2 Þ  Cq2 ðlk þ D þ jk Þ:  1 K s2 ; l k þ D þ jk

ð25Þ

where ReðsÞ > 0, ReðDÞ > 0. Proof. We will only give the proof of (23), because the proof of (24) and (25) is the same. We put

f ðtÞ ¼ t2D2

2. Illustrative Examples

m Y ð1Þ J 2lk ðak t; q2 Þ: k¼1

into the definition (14) yields

ð26Þ

j 1  n o X a2 H ðq2 Þ ð1Þ 1 j  Cq2 ðl þ D þ jÞ; qL2 t2D2 J2l ðat; q2 Þ; s ¼ Bðs; 2Þ  2 4s K ;lþ Dþ j 2 s j¼0  j  2 jðjþ22 lÞ 1  n o X q Hj ðq2 Þ a2 ð2Þ 1  Cq2 ðl þ D þ jÞ; qL2 t2D2 J2l ðat; q2 Þ; s ¼ Bðs; 2Þ  2 4s K s2 ; l þ D þ j j¼0

j 1  n o X a2 q2 ð3Þ qL2 t 2D2 J 2l ðat; q2 Þ; s ¼ Bðs; 1Þ  2 s j¼0  2 jðj1Þ q 2 Hj ðq2 Þ  Cq2 ðl þ D þ jÞ:  1 K s2 ; l þ D þ j

Please cite this article in press as: Al-Khairy, R.T. Journal of King Saud University – Science (2018), https://doi.org/10.1016/j.jksus.2018.08.012

ð27Þ

4

R.T. Al-Khairy / Journal of King Saud University – Science xxx (2018) xxx–xxx

Corollary 4. If one takes D ¼ nþ2 and 2 tively, one has

Bðs; sÞ ¼

an ½22nðs1Þ s2nþ2

l ¼ n2 in corollary (3), respec-

3. Concluding Remarks

;

The results proved in this paper give some contributions to the theory of the q-series especially q2 -Bessel functions. The above theorems and their various corollaries and consequences can be applied with different parameters to get different types of qLaplace transforms of wide range q2 -Bessel and q-Bessel functions. Also, the results obtained here can be used in some q-difference equations.

nþj

Hj ðqÞ ¼

ð1  qÞ 1 ¼ : ðq; qÞjþn ðq; qÞj ðq; qÞj Cq ðn þ j þ 1Þ

consequently

n

o

qL2 t n J nð1Þ ðat; q2 Þ; s ¼ n o qL2 t n J nð2Þ ðat; q2 Þ; s ¼

qL2

n

t n J nð3Þ ðat; q2 Þ; s

o

qL2

qL2

n

n

o

2 tn J ð2Þ n ðat; q Þ; s

o

2 tn J ð3Þ n ðat; q Þ; s

 

 a n 2 ½2s2nþ2

 a n 2

a2 4s2

eq2

;

Acknowledgment

 2 jðjþ22 lÞ  a2  j 1 q  4s2 X

½2s2nþ2

ðq2 ; q2 Þj

j¼0

The author would like to thank Professor S. K. Omari for his valuable suggestions.

;

  q2 a2 ðaÞn s2 ¼ E ; ½2s2nþ2 q2

n o 2 qL2 tn J ð1Þ n ðat; q Þ; s ¼

an 2 ½2s2nþ2

an ¼

which is the same result given in Yürekli and Sadek, 1991.

2 ½2s2nþ2

1 X j¼0

1 X j¼0

References



2

j

 4sa 2  ; ðq2 ; q2 Þj K s12 ; n þ j þ 1  j  2 jðjþnÞ 2 q 2  4sa 2  ; ðq2 ; q2 Þj K s12 ; n þ j þ 1

 jðj1Þ  2 2  j 1 ð1Þ j q2 2 a s2q an X  : ¼ ½2s2nþ2 j¼0 ðq2 ; q2 Þj K s12 ; n þ j þ 1

ð28Þ

Corollary 5. In corollary (4) if we use the relations (15), (16) and pffiffiffi replace a with 2 a we obtain the results of Purohit and Kalla (2007), for example in the first equation of corollary (4) n nn pffiffiffiffiffi o a2 ðasÞ Lq t 2 J ð1Þ ð2 at Þ; s ¼ eq : n snþ1

ð29Þ

Exton, H., Srivastava, H.M., 1994. A certain class of q-Bessel polynomials. Math. Comput. Model. 19 (2), 55–60. Hahn, W., 1949. Beitrage Zur Theorie Der Heineschen Reihen, die 24 Integrale der hypergeometrischen q-diferenzengleichung, das q-Analog on der Laplace transformation. Mathematische Nachrichten 2, 340–379. Ismail, M.E.H., 1981. The basic Bessel functions and polynomials. SIAM J. Math. Anal. 12 (3), 454–468. Ismail, M.E.H., 1982. The zeros of basic Bessel functions, the functions Jv + ax(x), and associated orthogonal polynomials. J. Math. Anal. Appl. 86 (1), 1–19. Jackson, F.H., 1905. The application of basic numbers to Bessel’s and Legendre’s functions. Proc. London Math. Soc. 2 (1), 192–220. Kac, V.G., De Sole, A., 2005. On Integral representations of q-gamma and q-beta functions. Accademia Nazionale dei Lincei (9), 11–29. Omari, S.K., 2017. On q-analogues of the natural transform of certain q-Bessel Functions and some application. Filomat 31 (9), 2587–2598. Purohit, S.D., Kalla, S.L., 2007. On q-Laplace transforms of the q-Bessel functions. Fract. Calc. Appl. Anal. 10 (2), 189–196. Uçar, F., 2014. q-Sumudu transforms of q-analogues of Bessel functions. Sci. World J. 2014, 1–7. Uçar, F., Albayrak, D., 2011. On q-Laplace type integral operators and their applications. J. Diff. Equ. Appl., 1–14 Yürekli, O., 1999a. Theorems on L2 -transforms and its applications. Complex Var. Elliptic Equ. 38, 95–107. Yürekli, O., 1999b. New identities involving the Laplace and the L2 -transforms and their applications. Appl. Math. Comput. 99, 141–151. Yürekli, O., Sadek, I., 1991. A Parseval-Goldstein type theorem on the Widder potential transform and its applications. Int. J. Math. Math. Sci. 14, 517–524.

which is the same result given by Purohit and Kalla (2007). Corollary 6. In corollary (5) if we take the limit as q ! 1

n o n o L2 t n J nð1Þ ðatÞ; s ¼L2 t n J nð1Þ ðatÞ; s  a n  2  ¼

2 2s2nþ2

e

a 4s2

:

ð30Þ

Please cite this article in press as: Al-Khairy, R.T. Journal of King Saud University – Science (2018), https://doi.org/10.1016/j.jksus.2018.08.012