ON q-integral TRANSFORMS AND THEIR APPLICATIONS

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1 Bulletin of Mathematical Analysis and Applications ISSN: 82-29, URL: Volume 4 Issue 2 22), Pages 3-5 ON q-integral TRANSFORMS AND THEIR APPLICATIONS COMMUNICATED BY R.K. RAINA) DURMUŞ ALBAYRAK, SUNIL DUTT PUROHIT AND FARUK UÇAR Abstract. In this paper, we introduce a q-analogue of the P-Widder transform and give a Parseval-Goldstein type theorem. Furthermore, we evaluate the q-laplace transform of a product of q-bessel functions. Several special cases of our results are also pointed out.. Introduction, Definitions and Preliminaries In the classical analysis, a Parseval-Goldstein type theorem involving the Laplace transform, the Fourier transform, the Stieltjes transform, the Glasser transform, the Mellin transform, the Hankel transform, the Widder-Potential transform and their applications are used widely in several branches of Engineering and applied Mathematics. Some integral transforms in the classical analysis have their q-analogues in the theory of q-calculus. This has led various workers in the field of q-theory for extending all the important results involving the classical analysis to their q- analouges. With this objective in mind, this paper introduces q-analogue of the P-Widder potential transform and establishes certain interesting properties for this integral transform. Throughout this paper, we will assume that q satisfies the condition < q <. The q-derivative D q f of an arbitrary function f is given by D q f)x) = where x. Clearly, if f is differentiable, then fx) fqx) q)x, lim D qf)x) = dfx) q dx. 2 Mathematics Subject Classification. Primary 5A3, 33D5; Secondary 44A,44A2. Key words and phrases. q-calculus, q-laplace transform, q-analogue of P-Widder transform and q-bessel function. c 22 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted October 5, 2. Accepted May, 22. 3

2 4 DURMUŞ ALBAYRAK, SUNIL DUTT PUROHIT AND FARUK UÇAR Before we continue, let us introduce some notation that is used in the remainder of the paper. For any real number α, [α] := qα q. In particular, if n Z +, we denote [n] = qn q = qn + + q +. Following usual notation are very useful in the theory of q-calculus: a; q) n = n k= ) aq k, a; q) = ) aq k, a; q) t = a; q) aq t t R). ; q) It is well known that in the literature there are two types of the q-laplace transform and studied in details by many authors. Hahn [4] defined q-analogues of the wellknown classical Laplace transform φs) = by means of the following q-integrals: L q ft); s} = q L s ft)} = q k= exp st)ft)dt Rs) > ), ) s E q qst)ft)d q t Rs) > ), 2) and L q ft); s} = q L s ft)} = e q st)ft)d q t Rs) > ), 3) q where the q-analogues of the classical exponential functions are defined by and E q t) = e q t) = n= n= t n = t < ), 4) q; q) n t; q) ) n q nn )/2 t n By virtue of the q-integral see [6,, 3]) q; q) n = t; q) t C). 5) ft)d q t = q) k= q k fq k ), 6) the q-laplace operator 3) can be expressed as ql s ft)} = q k s; q) k fq k ). 7) s; q) k Z Throughout this paper, we will use L q instead of q L s. The improper integral see [8] and [6]) is defined by /A fx)d q x = q) k Z q k A fqk ). 8) A

3 ON q-integral TRANSFORMS AND THEIR APPLICATIONS 5 The q-analogue of the integration theorem by a change of variable can be started when u x) = αx β, α C and β >, as follows: ub) ua) f u) d q u = b a f u x)) D q /β u x) d q /β x. 9) As a special cases of the formula 9), one has the following reciprocity relations: A /A fx)d q x = /A q/a A x 2 f x )d qx, fx)d q x = x 2 f x )d qx. ) Furthermore, the q-hypergeometric functions and well-known q-special functions are defined by see [9] and [7]): [ ] a a 2 a r rφ s ; q, z = b b 2 b s rψ s [ a a 2 a r b b 2 b s ; q, z and Γ q α) = / q) Γ q α) = K A; α) B q t; s) = Γ q t) Γ q s) Γ q t + s) a ; q) n a 2; q) n a r; q) n n= q; q) n b ; q) n b s; q) n ] = a ; q) n a r; q) n n= b ; q) n b s; q) n [ ) n q n 2)] +s r z n, [ ) n q n 2 )] s r z n x α E q q q) x) d q x α > ), ) /A q) x α e q q) x) d q x α > ), 2) t, s R), 3) where last two representations are based on the following remarkable function see [7, p.5]) K A; t) = A t q/a ; q) A ; q) q t /A ; q) Aq t t R). 4) ; q) Recently Uçar and Albayrak [2] introduced q-analogues of the L 2 -transfom in terms of the following q-integrals: and ql 2 f t) ; s} = /s q 2 t E q 2q 2 s 2 t 2 )ft)d q t Rs) > ), 5) ql 2 f t) ; s} = q 2 t e q 2 s 2 t 2 )ft)d q t Rs) > ). 6) In the same article [2], the author established an interesting relation between q L 2 and L q transforms, namely ql 2 f t) ; s} = [2] L q 2 f t /2) ; s 2}. 7) The paper is organized in the following manner. In the next two sections we introduce a q-analogue of the P-Widder potential transform and establish a Parseval- Goldstein type theorem and its corollaries involving q-analogue of the P-Widder and L 2 -Laplace transforms. Whereas in Section 4, we evaluate the q-laplace transform of a product of basic analogue of the Bessel functions. Several special cases

4 6 DURMUŞ ALBAYRAK, SUNIL DUTT PUROHIT AND FARUK UÇAR and examples of our results are also pointed out in the concluding section. 2. q-analogue of The P-Widder Transform Widder [3] has presented a theory of the integral transform P f x) ; y} = xf x) x 2 + y 2 dx, in the real domain which is formally equivalent to the iterated L 2 transform. Definition [2] A function f is q-integrable on [, ) if the series n Z qn f q n ) converges absolutely. We write L qr q ) for the set of all functions that are absolutely q-integrable on [, ), where R q is the set R q = q n : n Z}, that is } L q R q ) := := f : n Z q n f q n ) < f : q } f x) d q x <. Now we introduce the following q-integral transform, which may be regarded as q-extension of the P-Widder potential transform. Definition 2.. A q-analogue of P-Widder potential transform will be denoted P q and defined by the following q-integral: P q f x) ; s} = q 2 In view of 8), 8) can be expressed as P q f x) ; s} = + q x s 2 + q 2 x 2 f x) d qx, Rs) > ). 8) n Z q 2n s 2 + q 2+2n f qn ). 9) Now we prove the following theorem that provide the existence and convergence for the P q -transform: Theorem 2.. If f L q R q ), then the improper q-integral defined by 8) is well-defined. Proof. From 9), we have P q f x) ; s} q 2n + q s 2 + q 2+2n f qn ). n Z q n } Since s 2 + q 2+2n : n Z is bounded, then there exists a K R + such that q n s 2 + q 2+2n < K. Thus, we have P q f x) ; s} K q n f q n ). + q n Z

5 ON q-integral TRANSFORMS AND THEIR APPLICATIONS 7 On the other hand, since f L q R q ), there is a M R + such that q) f x) d q x = n Z q n f q n ) = M <. Hence we have This completes to proof. P q f x) ; s} MK + q <. 3. Main Theorems and Applications Proposition 3.. For A, t R, we have lim K A; t) =. q Proof. Multiplying numerator and denominator of K A; t) by + /A), we get K A; t) = A t + /A) q/a; q) A; q) + /A) q t /A; q) q t A; q) = A t /A; q) qa; q) q t /A; q) q t. A; q) and then using the formula as q see [3, p. 9,.3.8]) aq t ; q) a; q) into last expression, we obtain = φ q t ; ; q, a ) F t; ; a) = a) t, a <, t real lim K A; t) = lim A t /A; q) qa; q) q q q t /A; q) q t A; q) = lim A t q φ q t ; ; q, /A) φ q t ; ; q, qa) = A t + ) t + A) t A =. This completes to proof. Theorem 3.. The P q -Widder transform can be regarded as iterated q L 2 -Laplace transforms as under: ql 2 q L 2 f x) ; s} ; t} = [2] P q f x) ; t}, 2) provided that the q-integrals involved converge absolutely. Proof. On using the definition of the q L 2 -Laplace transform 6) the left-hand side of 2) say I) reduces to I = q 2 ) 2 se q 2 t 2 s 2) xe q 2 s 2 x 2) ) f x) d q x d q s. 2)

6 8 DURMUŞ ALBAYRAK, SUNIL DUTT PUROHIT AND FARUK UÇAR Interchanging the order of the q-integration in 2), which is permissible by the absolute convergence of q-integrals, we obtain I = q 2 ) 2 xf x) se q 2 t 2 + q 2 x 2) s 2) ) d q s d q x. In view of the definition 6), we obtain I = q 2 ql 2 ; t 2 + q 2 x 2) /2 } xf x) d q x. On setting fx) = and s = t 2 + q 2 x 2) /2 in 7) and then using the formula L q 2; s 2 } =, we get the desired result s2 I = [2] q 2 = [2] P q f x) ; t}. xf x) t 2 + q 2 x 2 d qx Corollary 3.. If 2 < α < then the following formula holds: P q x α ; t} = B q2 + α/2; α/2) [2] q 2 K ; + α/2) K /t 2 ; α/2) tα, 22) where KA; t) is given by 4). Proof. On setting fx) = x α and make use of the known result due to Uçar and Albayrak [2], namely ql 2 x α ; s} = Γ q 2 + α/2) ) q 2 α/2 [2] K /s 2 ; + α/2) s α+2 2 < α < ), the identity 2) of Theorem 3. give rise to P q x α Γ q 2 + α/2) ; t} = [2] ql ) }} q 2 α/2 2 [2] K /s 2 ; + α/2) s α+2 ; t = Γ q 2 + α/2) q 2) α/2 s α 2 } ql 2 K /s 2 ; + α/2) ; t. Using the series representation of the q L 2 -transform, we obtain P q x α ; t} = Γ q 2 + α/2) q 2) α/2 [2] t 2 ; q 2 ) n Z q 2n q n ) α 2 t 2 ; q 2) n K /q 2n. ; + α/2) Following Kac and Sole [7] the function of x, Kx; t) is a q-constant, that is, Kq n x; t) = Kx; t) for every integer n. Hence P q x α ; t} = Γ q 2 + α/2) q 2) α/2 K ; + α/2) [2] t 2 ; q 2 q 2n q n ) α 2 t 2 ; q 2) ). n n Z

7 ON q-integral TRANSFORMS AND THEIR APPLICATIONS 9 Again, on using the series representation of the q L 2 -transform, we get P q x α ; t} = Γ q 2 + α/2) q 2) α/2 K ; + α/2) = Γ q 2 + α/2) q 2) α/2 K ; + α/2) ql 2 s α 2 ; t } Γ q 2 α/2) q 2) α/2 [2] K /t 2 ; α/2) t α = Γ q2 + α/2) Γ q 2 α/2) [2] q 2 K ; + α/2) K /t 2 ; α/2) tα = B q2 + α/2; α/2) [2] q 2 K ; + α/2) K /t 2 ; α/2) tα. In the following theorem, we establish a Parseval-Goldstein type theorem that involving q-analogue of the P-Widder and L 2 -Laplace transforms: Theorem 3.2. If P q and q L 2 denote q-analogues of the P-Widder and L 2 -Laplace transforms, then the following result holds true: x q L 2 f y) ; x} q L 2 g z) ; x} d q x = [2] provided that the q-integrals involved converge absolutely. yf y) P q g z) ; qy} d q y, 23) Proof. Using the definition of the q L 2 -transform 6), the left-hand side of 23) say J) yields to J = q 2 x q L 2 g z) ; x} ye q 2 x 2 y 2) } f y) d q y d q x. Changing the order of the q-integration, which is permissible by the hypothesis, we find that J = q 2 yf y) xe q 2 q 2 y 2 x 2) } ql 2 g z) ; x} d q x d q y. In view of the definition 6) and the result 2) of Theorem 3., the above relation reduces to the desired right-hand side of 23). Corollary 3.2. We have xh x) q L 2 f y) ; x} d q x = provided that the q-integrals involved converge absolutely. yf y) q L 2 h x) ; qy} d q y, 24) Proof. The identity 24) follows immediately after letting h x) = q L 2 g z) ; x} in the relation 23). Corollary 3.3. With due regards to convergence, we have q 2 P q ql 2 g u) ; x} ; q 2 z } = q L 2 P q g u) ; qy} ; qz}. 25)

8 DURMUŞ ALBAYRAK, SUNIL DUTT PUROHIT AND FARUK UÇAR Proof. To prove 25), we set f y) = e q 2 y 2 z 2), then we get ql 2 f y) ; x} = q 2 ye q 2 x 2 y 2) e q 2 q 2 z 2 y 2) d q y = [2] x 2 + q 2 z 2. 26) Substituting 26) into the identity 23) of Theorem 3.2, we obtain x x 2 + q 2 z 2 q L 2 g u) ; x} d q x = ye q 2 q 2 z 2 y 2) P q g u) ; qy} d q y q 2 q 2 P q ql 2 g u) ; x} ; q 2 z } = q L 2 P q g u) ; qy} ; qz}. Similarly, if we set f y) = E /q 2 y2 z 2 ) in Theorem 3.2 and using the wellknown q-exponential identity e q 2q 2 x) = E /q 2x) we find that ql 2 E /q 2 y2 z 2 ) } q 2 ; x = q L 2 eq 2 q 2 z 2 y 2) ; x } = [2] x 2 + q 2 z 2 27) Substituting 27) into 23) we obtain x x 2 + q 2 z 2 q L 2 g u) ; x} d q x = and finally we have q 2 P q ql 2 g u) ; x} ; q 2 z } = Theorem 3.3. We have ye /q 2 z 2 y 2) P q g u) ; qy} d q y ye q 2 q 2 z 2 y 2) P q g u) ; qy} d q y = q L 2 P q g u) ; qy} ; qz}. x q L 2 h y) ; x 2 + q 2 z 2) /2 } ql 2 g z) ; x} d q x = q2 ql 2 h y) P q g z) ; qy} ; qz} 28) [2] provided that the q-integrals involved converge absolutely. Proof. Let f y) = e q 2 y 2 z 2) h y). Using the definition of the q L 2 -transform we obtain ql 2 eq 2 y 2 z 2) h y) ; x } = q 2 ye q 2 x 2 + q 2 z 2) y 2) h y) d q y Substituting 29) into 23), we find that = [2] = q L 2 h y) ; x 2 + q 2 z 2) /2 }. 29) x q L 2 h y) ; x 2 + q 2 z 2) /2 } ql 2 g z) ; x} d q x = q2 [2] ye q 2 q 2 z 2 y 2) h y) P q g z) ; qy} d q y ql 2 h y) P q g z) ; qy} ; qz}.

9 ON q-integral TRANSFORMS AND THEIR APPLICATIONS It is interesting to observe that, if we set h y) = and make use of the Theorem 3.3, one can easily deduced Corollary 3.3. Corollary 3.4. The following result holds true: ql 2 s 2 q L 2 f x) ; } } ; t = s [2] P q f x) ; t}, 3) provided that the q-integrals involved converge absolutely. Proof. In view of the definition of the q L 2 -transform, the left-hand side of 3) say L) yields to L = q 2 s e q t 2 s ) s q L 2 f x) ; } d q s s = q 2 ) 2 s 3 e q t 2 s ) 2 2 xe q 2 s ) ) 2 x2 f x) d q x d q s = q 2 ) 2 xf x) s 3 e q 2 t 2 s 2 ) e q 2 s ) ) 2 x2 d q s d q x. On making use of the identity ) into the right-hand side, we obtain L = q 2 ) 2 xf x) se q 2 t 2 + q 2 x 2) s 2) ) d q s d q x = q 2 = [2] q 2 = [2] P q f x) ; t}. xf x) q L 2 ; t 2 + q 2 x 2) /2 } d q x xf x) t 2 + q 2 x 2 d qx 4. q-laplace Image of a Product of q-bessel functions Recently, Purohit and Kalla [] evaluated the q-laplace image under the L q operator 2) for a product of basic analogue of the Bessel functions. In this section, we propose to add one more dimension to this study by introducing a theorem which give rise to q-laplace image under the L q operator 3) for a product of q- Bessel functions. The third q-bessel function is defined by Jackson and in some literature it is called Hahn-Exton q-bessel function. For further details see [5]. J 3) ν t; q) = q ν+ ; q ) [ t ν Φ q; q) ] q ν+ ; q, qt 2 = t ν ) n q nn )/2 qt 2) n. q; q) ν+n q; q) n n= 3)

10 2 DURMUŞ ALBAYRAK, SUNIL DUTT PUROHIT AND FARUK UÇAR Theorem 4.. Let J 3) 2µ j a j t; q), j =, 2,..., n be n different q-bessel functions. Then, q-laplace transform of their product is as follow s υ M L q f t) ; s} = q)υ M Γ q υ + M) a µ...aµn n 32) Γ q 2µ + )...Γ q 2µ n + ) K /s; υ + M) q υ+m ; q ) m a /s) m... a n /s) mn q 2µ+ ; q) m... q 2µn+... ; q) mn q; q) m q; q) mn q. mm )/2 m,...,m n= ) ) where ft) = t υ q µ J 3) 2µ q v+m a t; q q µn J 3) 2µ n q v+m a n t; q, M = µ + + µ n, Re s) >, Re υ + M) >. Proof. To prove the above theorem we put ) ) f t) = t υ q µ J 3) 2µ q v+m a t; q q µn J 3) 2µ n q v+m a n t; q, into 7) and make use of 3), to obtain M = µ + + µ n L q f t) ; s} = q j s; q) s; q) j q j ) υ j Z q 2µ + ; q ) q; q) q v+m a q j) µ m,,m n= q 2µ n+ ; q ) q v+m a n q j) µ n q; q) q mm )/2 q v+m a q j) m q 2µ+ qmnmn )/2 q v+m a n q j) m n ; q) m q; q) m q 2µn+. ; q) mn q; q) mn On interchanging the order of summations, which is valid under the conditions given with theorem, we obtain q 2µ + ; q ) q 2µ n+ ; q ) L q f t) ; s} = a µ s; q) q; q) q; q) aµn n q v+m ) M q ) mm )/2 q v+m m ) a q 2µ+ qmnmn )/2 q v+m mn a n ; q) m q; q) m q 2µn+. ; q) mn q; q) mn m,,m n= s; q) j q jυ+µ+ +µn+m+ +mn) j Z } By using the well-known q-gamma function Γ q t) = q; q) q t ; q) q) t, and then summing the inner series with the help of the bilateral summation formula see [3, p. 26, 5.2.]), namely ψ b; c; q, z) = n Z b; q) n c; q) n z n = q, c/b, bz, q/bz; q) c, q/b, z, c/bz; q) c/b < z < ),

11 ON q-integral TRANSFORMS AND THEIR APPLICATIONS 3 we have L q f t) ; s} = s; q) m,,m n= We may rewrite this series q) 2M Γ q 2µ + ) Γ q 2µ n + ) aµ aµn n a ) m q 2µ+ ; q) m q; q) m a n ) mn q 2µn+ ; q) mn q; q) mn q,, sq υ+m+m + +m n, q υ+m+m+ +mn) /s; q ), q/s, q υ+m+m+ +mn, ; q), q) 2M L q f t) ; s} = Γ q 2µ + ) Γ q 2µ n + ) aµ aµn n q v+m ) M q ) mm )/2 q v+m m ) a q 2µ+ qmnmn )/2 q v+m mn a n ; q) m q; q) m q 2µn+ ; q) mn q; q) mn m,,m n= q; q) sq υ+m+m ; q ) q υ+m+m) /s; q ) q υ+m+m, 33) ; q) s; q) q/s; q) where m = m + + m n. Setting A = s and t = υ + M + m in 4), we get ) υ+m+m s s; q) K /s; υ + M + m) = /s; q) q s + s q υ+m+m s; q) υ+m+m) /s; q ) ) υ+m+m s; q) = q/s; q) q s q υ+m+m s; q) υ+m+m) /s; q ). 34) Substituting relation 34) into 33), we obtain q) 2M L q f t) ; s} = Γ q 2µ + ) Γ q 2µ n + ) aµ aµn n q v+m ) M q ) mm )/2 q v+m m ) a q 2µ+ qmnmn )/2 q v+m mn a n ; q) m q; q) m q 2µn+ ; q) mn q; q) mn m,,m n= q; q) q υ+m+m ; q) K /s; υ + M + m). s υ+m+m. Finally, on considering the following remarkable identity q t+m ; q ) = qt ; q) q t m N), ; q) m we have K t; s) = q s K t; s ) q) υ M a µ L q f t) ; s} = aµn n Γ q υ + M) Γ q 2µ + ) Γ q 2µ n + ) K /s; υ + M) s υ+m m,,m n= This completes the proof. q υ+m ; q ) m q 2µ+ ; q) m q 2µn+ ; q) mn a /s) m... a n /s) mn q; q) m q; q) mn q mm )/2.

12 4 DURMUŞ ALBAYRAK, SUNIL DUTT PUROHIT AND FARUK UÇAR 5. Special Cases In this section, we briefly consider some consequences and special cases of the results derived in the preceding sections. If we take n =, µ = υ, υ = µ and a = a in 32), we obtain } L q t µ q v J 3) q v+µ at; q) ; s 2v = qµ+υ ) v Γ q µ + υ) q) µ v s υ+µ Γ q 2v + ) K /s; µ + υ) av 2Φ q µ+υ, q 2v+ ; q, a s where Re µ + υ) > and Re s) >. Again, if we write υ 2 + and υ 2 instead of µ and υ in 35), respectively, we obtain L q qt) υ/2 J υ 3) qv at; q ) ; s} = aυ/2 s υ q vv+)/2 e q a/s) 36) K /s; υ + ) Now, setting υ = in 36) we obtain L q qt) /2 J 3) ) } qat; q ; s = a /2 s 2 e q a/s) Re s) > ). 37) Similarly, if we set υ = in 36), then we have ) } L q J 3) at; q ; s = s e q a/s) Re s) > ). 38) In 35) we write υ = and then a =, we find that ) 35) L q t µ ; s } = Γ q µ) q) µ s µ K /s; µ). 39) If we let q, and make use of the limit formulae and lim Γ q t) = Γ t), q lim q q a ; q) n q) n = a) n lim K A; t) = q where a) n = a a + )... a + n ), we observe that the identity 28) of Theorem 3.3 and 25) of Corollary 3.3 provide, respectively, the q-extensions of the known related results due to Yürekli [4, p. 97, Theorem and Corollary ]. Also, the results 32), 35), 36), 37) and 38) provide, respectively, the q-extensions of the following known results given in Erdélyi, Magnus, Oberhettinger and Tricomi [, pp ]: L t υ J 2µ 2 a t ) J 2µ2 2 a2 t ) J 2µn 2 an t ) ; s } a µ = aµn n Γ υ + M) Γ 2µ + ) Γ 2µ n + ) s υ+m Ψ n) 2 υ + M; 2µ +,, 2µ n + ; a s,, a ) n s where M = µ µ n, Re s) > and Re υ + M) >. L t µ J 2υ 2 ) } [ ] Γ µ + υ) aυ µ + υ; at ; s = s µ+υ Γ 2υ + ) F a/s, 2υ + ; L t υ/2 J υ 2 ) } at ; s = a υ/2 s υ e a/s,

13 and ON q-integral TRANSFORMS AND THEIR APPLICATIONS 5 where Re s) > and Re µ + υ) >. L t /2 J 2 ) } at ; s = a /2 s 2 e a/s, L J 2 ) } at ; s = s e a/s, Acknowledgement. The authors are thankful to the referee for his/her valuable comments and suggestions which have helped in improvement of the paper. The authors also thanks to Professor M.E.H.Ismail for sharing his unpublished work. References [] Erdélyi A., Magnus W., Oberhettinger F. and Tricomi F.G., Tables of Integral Transforms, Vol. I, McGraw-Hill Inc., New York, 954. [2] Fitouhi A., Bettaibi N. and Brahim K.,The Mellin transform in Quantum Calculus, Constr. Approx. 233) 26) [3] Gasper G. and Rahman M., Basic Hypergeometric Series, Cambridge University Press, Cambridge, 99. [4] Hahn W., Beitrage zur theorie der heineschen reihen, die 24 integrale der hypergeometrischen q-diferenzengleichung, das q-analog on der Laplace transformation, Math. Nachr., 2 949), [5] Ismail M.E.H., Some properties of Jackson s third q-bessel function, preprint. [6] Jackson F. H., On q-definite Integrals, Quart. J. Pure and Appl. Math. 4 9), [7] Kac V.G. and De Sole A., On integral representations of q-gamma and q-beta functions, Rend. Mat. Acc. Lincei 9 25), -29. [8] Koornwinder T. H., Special functions and q-commuting variables, in special functions, q- series and related topics Toronto,ON,995), volume 4 of Fields Inst. Commun., pp. 3-66, Amer.Math.Soc., Providence, RI, 997. [9] Koelink H.T. and Koornwinder T.H., Q-special functions, a tutorial, in deformation theory quantum groups with applications to mathematical physics, amherst, MA, 99) volume 34 of Contemp. math., pp. 4-42, Amer. Math. Soc., Providence, RI, 992. [] Purohit S.D. and Kalla S.L., On q-laplace transforms of the q-bessel functions, Fract. Calc. Appl. Anal., 2) 27), [] Thomae J., Beitrage zur theorie der durch die heinesche reihe., J. Reine angew.math, 7 869) [2] Uçar F. and Albayrak D., On q-laplace type integral operators and their Applications, J. Difference Equ. Appl. ifirst article, 2, -4. doi:.8/ [3] Widder D.V., A transform related to the Poisson integral for a half-plane, Duke Math. J ), [4] Yürekli O., Theorems on L 2 -transforms and its applications, Complex Variables Theory Appl ), Department of Mathematics, Faculty of Science and Letters, Marmara University, TR Kadıköy, Istanbul, Turkey, address: durmusalbayrak@marun.edu.tr, fucar@marmara.edu.tr Department of Basic Science Mathematics), College of Technology & Engineering, M.P. University of Agri. & Tech., Udaipur-33, India, address: sunil a purohit@yahoo.com