On Some q-Analogues of the Natural Transform and Further Investigations
aa r X i v : . [ m a t h . C A ] A p r On Some q-Analogues of the Natural Transform andFurther Investigations
S. K. Q. Al-Omari
Department of Applied Sciences; Faculty of Engineering Technology; Al-Balqa’Applied University; Amman 11134; [email protected]
Abstract
Some q -analogues of classical integral transforms have recently been investigated by manyauthors in diverse citations. The q -analogues of the Natural transform are not known norused. In the present paper, we are concerned with definitions and investigations of the q -theory of the Natural transform and some applications. We present two types of q -analoguesof the cited transform on given sets and get results of the nominated analogues for certainclass of functions of special type. We declare here that given results are new and theycomplement recent known results related to q -Laplace and q -Sumudu transforms. Over andabove, we present some supporting examples to illustrate effectiveness of the given results . Keywords : q -Sumudu transform; q -Laplace transform; q -Natural transform; q -Besselfunction. The study of q -analogues of classical integral transforms has not yet been developed to agreat extent. This partially can be explained by the fact that one is not very familiar withthe q -theory and that basic q -integral transforms do not occur frequently in physics. It by nomeans our aim to give in this paper a new general results in the theory of q -calculus. But werestrict our selves to the necessary theory to give some q -analogues of an integral transformnamed as the Natural transform and to estimate the transform relevant properties. Theclassical theory of the Natural transform is closely related to the classical theory of Laplaceand Sumudu transforms, two of the best known of all integral transforms. The Naturaltransform for functions of exponential order is defined over the set A,A = (cid:26) f ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ∃ M, τ and/or τ > | f ( t ) | < M e | t | τj , t ∈ ( − j × [0 , ∞ ) , j = 1 , (cid:27) (1)by the integral equation N ( f ) ( u ; v ) = Z ∞ f ( ut ) exp ( − vt ) dt, (2)where Re v > u ∈ ( − τ , τ ) , u and v being the transform variables.The Natural transform strictly converges to the Sumudu transform (See [28 , ,
30] ) for v = 1 and it strictly converges to the Laplace transform for u = 1 . Further fundamentalproperties of this transform and its application to differential equations are given by [14]and [15 , , . We organize this paper as follows. In section 2, we recall some definitions and notationsfrom the q -calculus. In Section 3, we specify the q -analogues of the Natural transformin terms of a series representation. In Section 4, we apply the first of two analogues of S.K.Q. Al-Omari the Natural transform to a certain class of special functions and pick up some results byconsidering q -Laplace and q -Sumudu transforms. In Section 5, we figure out some valuesof the second representation of the q -transform and extend the resulting theorems to thecase of q -Laplace and q -Sumudu transforms. In Section 6 and 7, we derive certain resultsconcerning Fox’s H q -function and propose some counter examples as an application to theprevious theory. q -Calculus For the convenience of the reader, we provide a summary of mathematical notations used inthis paper. Throughout this paper, wheresoever it appears , q statisfies the condition that0 < q < . The q -calculus begins with the definition of the q -analogue d q f ( x ) of the differential of thefunction, i.e., d q f ( x ) = f ( x ) − f ( qx ) , and the q -analogue of the derivative ,d q f ( x ) d q x = f ( x ) − f ( qx )( q − x . On certain additional requirements, the q -analogue may be unique, but sometimes it is usefulto consider several q -analogues of the same object.The q -analogues of an integer n ( q -integer), a factorial of n ( q -factorial of n ) , and the binomialcoefficient (cid:0) nk (cid:1) ( q -binomial coefficient) are respectively given as[ n ] q = 1 − q n − q , (cid:16) [ n ] q (cid:17) ! = n Q [ k ] q , n = 1 , , , ... , n = 0 and (cid:20) nk (cid:21) q = n Y − q n − k +1 − q k . Clearly, lim q → [ n ] q = n, lim q → (cid:16) [ n ] q (cid:17) ! = n ! and lim q → (cid:20) nk (cid:21) q = (cid:18) nk (cid:19) . If α ∈ C , then the q -analogue of α is given as 1 − q α − q and, it sometimes makes sense when α is not, [ ∞ ] q = 11 − q . If n ∈ N , then the q -analogue of ( x + a ) n and the derivative are respectively given as( x + a ) nq = n − Y j =0 (cid:0) x + q j a (cid:1) and D q ( x + a ) nq = [ n ] ( x + a ) n − q , ( x + a ) q = 1 . (3)If x = 1 and a = x, then the above formula makes sense for n = ∞ , giving(1 + x ) ∞ q = ∞ Y (cid:0) q k x (cid:1) . (4)The q -Jackson integral from 0 to a is given by Jackson [11] as Z a f ( x ) d q x = (1 − q ) a ∞ X f (cid:0) aq k (cid:1) q k , (5) n Some q-Analogues of the Natural Transform and ... q -Jackson integral in a generic interval [ a, b ] is given by [11] Z ab f ( x ) d q x = Z b f ( x ) d q x − Z a f ( x ) d q x. (6)The improper integral is defined as [5] Z ∞ A f ( x ) d q x = (1 − q ) X n ∈ Z q k A f (cid:18) q k A (cid:19) . (7)The q -analogues of the gamma function are defined by [5]Γ q ( α ) = Z − q x α − E q ( q (1 − q ) x ) d q x, q Γ ( α ) = K ( A ; α ) Z ∞ A (1 − q ) x α − e q ( − (1 − q ) x ) d q x, where α > K ( A ; t ) = A t − ( − q/A ; q ) ∞ ( − q t /A ; q ) ∞ ( − A ; q ) ∞ ( − Aq − t ; q ) ∞ , (8)( a ; q ) n = n − Y (cid:0) − aq k (cid:1) , ( a ; q ) ∞ = lim n →∞ ( a ; q ) n . (9)for all t ∈ R . The useful notations we need here are [10]Γ q ( x ) = ( q ; q ) ∞ ( q x ; q ) ∞ (1 − q ) − x and ( a ; q ) t = ( a ; q ) ∞ ( aq t ; q ) ∞ , t ∈ R . (10)The q -analogues of the exponential function are given as E q ( t ) = ∞ X ( − n q n ( n − ( q ; q ) n t n = ( t ; q ) ∞ , t ∈ C e q ( t ) = ∞ X q ; q ) n t n = 1( t, q ) ∞ , t < . (12)The q -analogues of the hypergeometric function are defined in two ways as r φ s (cid:20) a , a , ..., a r b , b , ..., b s (cid:12)(cid:12)(cid:12)(cid:12) q, z (cid:21) = ∞ X ( a , a , ..., a r ; q ) n ( b , b , ..., b s ; q ) n z n ( q ; q ) n (13)and m − k Φ m − (cid:20) a , a , ..., a m − k b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q, z (cid:21) = ∞ X ( a , ..., a m − k ; q ) n ( b , ..., b m − ; q ) n h ( − n q ( n ) i k z n ( q ; q ) n , (14)where ( a , a , ..., a p ; q ) n = p Q ( a k , q ) n . S.K.Q. Al-Omari
For | z | <
2, some q -analogues of the Bessel function are defined as J (1) v ( z ; q ) = (cid:0) q v +1 ; q (cid:1) ∞ ( q ; q ) ∞ (cid:16) z (cid:17) Φ (cid:20) q v +1 (cid:12)(cid:12)(cid:12)(cid:12) q, − z (cid:21) = (cid:16) z (cid:17) v ∞ X n =0 (cid:16) − z (cid:17) n ( q ; q ) v + n ( q ; q ) n , (15) J (2) v ( z ; q ) = ( q v +1 ; q ) ∞ ( q ; q ) ∞ (cid:0) z (cid:1) v Φ (cid:20) q v +1 (cid:12)(cid:12)(cid:12)(cid:12) q, − q v +1 z (cid:21) = (cid:0) z (cid:1) v ∞ X q n ( n + v ) ( q ; q ) v + n ( q ; q ) n (cid:18) − z (cid:19) n , (16) J (3) v ( z ; q ) = (cid:0) q v +1 ; q (cid:1) ∞ ( q ; q ) ∞ z v Φ (cid:20) q v +1 (cid:12)(cid:12)(cid:12)(cid:12) q, qz (cid:21) = z v ∞ X ( − n q n ( n − ( q ; q ) v + n ( q ; q ) n . (17) q -Analogues of the Natural Transform Theory and applications of q -integral transforms are evolving rapidly over the recent years.Since Jackson [11] presented a precise definition of so-called q -Jackson integral and developed q -calculus in a systematic way. It was well known that, in the literature, there are two typesof q -analogues of integral transforms studied in detail by many authors in the recent pastsuch as Abdi [2] , Hahn [18] , Purohit and Kalla [3] , Albayrak [25], U¸car and Albayrak [4] , Albayrak et al. [5] and [6] , Yadav and Purohit [7] , Fitouhi and Bettaibi [8] and [9] andmany others, to mention but a few.In this section of this paper we deem it proper to give the definition of the q -analoguesof the Natural transform as in the following definition. Definition 1.
Let ˆ A and ˇ A be defined by ˆ A = n f ( t ) (cid:12)(cid:12)(cid:12) ∃ M, τ and/or τ > | f ( t ) | < M E q (cid:16) | t | τ j (cid:17) , t ∈ ( − j × [0 , ∞ ) , j = 1 , o and ˇ A = n f ( t ) (cid:12)(cid:12)(cid:12) ∃ M, τ and/or τ > | f ( t ) | < M e q (cid:16) | t | τ j (cid:17) , t ∈ ( − j × [0 , ∞ ) , j = 1 , o , respectively. Then, we have the following definitions. ( i ) Over the set ˆ A, we define the q -analogue of the Natural transform of first type as N q ( f ) ( u ; v ) = 1(1 − q ) u Z uv f ( t ) E q (cid:16) q vu t (cid:17) d q t, (18) provided the integral exists. ( ii ) Over the set ˇ A, we define the q -analogue of the Natural transform of type two as q N ( f ) ( u ; v ) = 1(1 − q ) Z ∞ f ( t ) e q (cid:16) − vu t (cid:17) d q t, (19) when the integral exists. It seems very benificial to us to notice the following relations N q ( f ) (1; v ) = ( L q f ) ( v ) , q N ( f ) (1; v ) = ( q Lf ) ( v ) ,N q ( f ) ( u ; 1) = ( S q f ) ( u ) , q N ( f ) ( u ; 1) = ( q Sf ) ( u ) , (20)where L q ( S q ) and q L ( q S ) are respectively the q -analogues of the Laplace (Sumudu) trans-forms of first (second ) types; see, for example, [4] ( resp. , [6]). n Some q-Analogues of the Natural Transform and ... q -analogue of (18) can be expressedas N q ( f ) ( u ; v ) = ( q ; q ) ∞ v X k ≥ q k f (cid:16) uv q k (cid:17) ( q ; q ) k , (21)whereas, the q -analogue of (19) can similarly be performed in terms of that series as q N ( f ) ( u ; v ) = ∞ X k ∈ Z ( q ; q ) ∞ v f (cid:0) q k (cid:1)(cid:16) − uv q k ; q (cid:17) ∞ . Hence, on parity of the fact ( a ; q ) k = ( a ; q ) ∞ ( aq k ; q ) ∞ ( for a = − vu ) , the previous equation hasthe series representation q N ( f ) ( u ; v ) = 1 (cid:16) − vu ; q (cid:17) ∞ ∞ X k ∈ Z q k f (cid:0) q k (cid:1) (cid:16) − uv ; q (cid:17) k (22)that we shall use in later investigations. N q of Some Special Functions In this section of this paper, we apply the analogue N q for certain functions of special typeand extend the work to q -Laplace and q -Sumudu transforms. We assume the functions,unless otherwise stated, are of power series form, f ( x ) = X n ≥ A n x n , (23)where A n is some bounded sequence .In what follows, we establish the following three main theorems of this section. Theorem 1.
Let α be a positive real number and f ( x ) = P n ≥ A n x n . Then, we have N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (1 − q ) α − v α u α − X n ≥ A n u n v n (1 − q ) n Γ q ( α + n ) . (24) Proof
Let α be a positive real number and f ( x ) = P n ≥ A n x n be given. Then, on aid of (21) , we write N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = ( q ; q ) ∞ v X k ≥ q k (cid:16) uv q k (cid:17) α − f (cid:16) uv q k (cid:17) ( q ; q ) k = u α − v α ( q ; q ) ∞ X k ≥ q αk ( q ; q ) k X n ≥ A n (cid:16) uv q k (cid:17) n = u α − v α ( q ; q ) ∞ X n ≥ A n (cid:16) uv (cid:17) n X k ≥ q k ( α + n ) ( q ; q ) k . (25) S.K.Q. Al-Omari
Hence, taking into account (11) , (25) simply reveals N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = u α − v α ( q ; q ) ∞ X n ≥ A n (cid:16) uv (cid:17) n e q (cid:0) q α + n (cid:1) = u α − v α ( q ; q ) ∞ X n ≥ A n u n v n (cid:0) q ( α + n ) ; q (cid:1) ∞ . (26)Therefore, by aid of the Equations (10) and (26) we get that N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = u α − v α ( q ; q ) ∞ X n ≥ A n (cid:16) uv (cid:17) n e q (cid:0) q α + n (cid:1) = (1 − q ) α − v α u α − X n ≥ A n u n v n Γ q ( α + n )(1 − q ) − n = (1 − q ) α − v α u α − X n ≥ A n u n (1 − q ) n v n Γ q ( α + n ) . This completes the proof of the theorems.
Theorem 2.
Let α be a positive real number. Then, the following hold. ( i ) Let Γ q ( α ) be the q-gamma function of the first type . Then, we have N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (1 − q ) α − v α u α − Γ q ( α ) . ( ii ) Let a ∈ R and f ( x ) = m − k Φ m − (cid:20) a , a , ..., a m − k b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q, ax (cid:21) . Then, we have (cid:0) N q x α − f ( x ) (cid:1) ( u ; v ) = Γ q ( α ) (1 − q ) α − u α − v αm − k +1 Φ m (cid:20) a , a , ..., a m − k q α b , b , ..., b m − , (cid:12)(cid:12)(cid:12)(cid:12) q, auv (cid:21) . (27) Proof of ( i ) By assuming A = 1 and A n = 0 , ∀ n ≥ , it follows from (23) that f ( x ) = 1 . Hence, the first part obviously follows.
Proof of ( ii ) Appealing to the q -analogue (14) , f ( x ) can fairly be written as f ( x ) = X n ≥ ( a , a , ..., a m − k ; q ) n ( b , b , ..., b m − ; q ) n (cid:16) ( − n q n ( n − (cid:17) q n ( q ; q ) n x n . On setting A n = ( a , a , ..., a m − k ; q ) n ( b , b , ..., b m − ; q ) n (cid:16) ( − n q n ( n − (cid:17) k a n , (28)we get the power series representation of type (23) . Hence, Theorem 1 reveals N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (1 − q ) α − v α u α − X n ≥ A n u n v n (1 − q ) n Γ q ( α + n ) . Therefore, on using the identity of the gamma function [1]Γ q ( x + j ) = ( q x ; q ) j (1 − q ) j Γ q ( x ) , (29) n Some q-Analogues of the Natural Transform and ... N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (1 − q ) α − v α u α − X n ≥ A n u n v n ( q α ; q ) n Γ q ( α )= Γ q ( α ) (1 − q ) α − u α − v α X n ≥ A n u n v n ( q α ; q ) n u n v n = Γ q ( α ) (1 − q ) α − u α − v α m − k +1 Φ m (cid:20) a , a , ..., a m − k q α b , b , ..., b m − , (cid:12)(cid:12)(cid:12)(cid:12) q, auv (cid:21) . This completes the proof of the theorem.An inspection of the previous two main theorems leads to the following list of inclusions:On setting k = m = 1 , Theorem 2 ( ii ) (for α = 1) yields N q ( E q ( ax )) ( u ; v ) = 1 v Φ (cid:20) q (cid:12)(cid:12)(cid:12)(cid:12) q, auv (cid:21) . (30)Similarly, by inserting α = 1, Theorem 2 ( i ) instantly shows( N q (1)) ( u ; v ) = 1 v . (31)Moreover, by setting α = n + 1 in the first part of the theorem , it yields N q ( x n ) ( u ; v ) = (1 − q ) n v n +1 u n (cid:16) [ n ] q (cid:17) ! . (32)Therefore, (31) and (32) when designated reveal N q (cid:0) x + ... + x n (cid:1) ( u ; v ) = X k ≥ (1 − q ) k v k +1 u k (cid:16) [ k ] q (cid:17) ! . We finally establish the main third theorem of this section.
Theorem 3.
Let f ( x ) be given as f ( x ) = r φ p (cid:20) a , a , ..., a r b , b , ..., b p (cid:12)(cid:12)(cid:12)(cid:12) q, ax (cid:21) and α > . Then, wehave N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = Γ q ( α ) (1 − q ) α − u α − v α r +1 φ p (cid:20) a , a , ..., a r , q α b , b , ..., b p (cid:12)(cid:12)(cid:12)(cid:12) q, a uv (cid:21) . (33) Proof
By taking into account (13) , f ( x ) is given the series representation f ( x ) = X n ≥ ( a , a , ..., a m − k ; q ) n ( b , b , ..., b m − , q ) n a n ( q ; q ) n x n . On setting A n = ( a , a , ..., a m − k ; q ) n ( b , b , ..., b m − , q ) n a n ( q ; q ) n and using (21) and (29) , we get N q (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (1 − q ) α − u α − v α X n ≥ A n u n v n (1 − q ) n Γ q ( α + n )= Γ q ( α ) (1 − q ) α − u α − v α X n ≥ A n ( q α ; q ) n u n v n = Γ q ( α ) (1 − q ) α − u α − v α r +1 φ ρ (cid:20) a , a , ..., a r , q α b , b , ..., b p (cid:12)(cid:12)(cid:12)(cid:12) q, a uv (cid:21) . S.K.Q. Al-Omari
This completes the proof of the theorem.By setting p = 0 , α = 1 and r = 0 in (33), the above theorem leads to N q ( e q ) ( ax ) ( u ; v ) = 1 v φ h q ; − (cid:12)(cid:12)(cid:12) q, a uv i = 1 v X n ≥ (cid:16) auv (cid:17) n = 1 v − au , | au | < v. (34)Further, by fixing v = 1 and taking account of (20) , (34) spreads the result to the case of q -Sumudu transform giving S q ( e q ) ( ax ) ( u ) = 11 − au , | au | < . Similarly, by fixing u = 1 and consulting (20) yield the following case of q -Laplace transform L q ( e q ( ax )) ( v ) = L q ( e q ( ax )) ( v ) = 1 v − a , | a | < v. From above investigations, we, further, deduce N q (sin q ax ) ( u ; v ) = N q (cid:18) e q ( iax ) − e q ( − iax )2 i (cid:19) ( u ; v ) = auv + a u , | au | < v, (35)and N q (cos q ax ) ( u ; v ) = N q (cid:18) e q ( iax ) + e q ( − iax )2 i (cid:19) ( u ; v ) = vv + a u , | au | < v. (36)Hence, by virtue of (35) and (36) we state without proof the following corrollary. Corrollary 4.
Let a be a real number. Then, the following hold. (i) S q (sin q ax ) ( u ) = au a u , | au | <
1; (ii) S q (cos q ax ) ( u ) = 11 + a u , | au | < , (iii) L q (sin q ax ) ( v ) = av + a , | a | < v ; (iv) L q (cos q ax ) ( v ) = vv + a , | a | < v. From Theorem 1 we state and prove the following corrollary.
Corrollary 5.
Let a be a real number and f ( x ) = X n ≥ A n x n . Then, we have N q (cid:16) x α − J (1)2 µ (2 √ ax ; q ) (cid:17) ( u ; v ) = (1 − q ) α − u α − v α Γ q ( α ) X n ≥ A n u n v n ( q α ; q ) n . Proof
Let a be a real number, then by aid of (15) , we consider to write J (1)2 µ (cid:0) √ ax ; q (cid:1) = x µ X n ≥ ( − n a µ + n ( q ; q ) µ + n ( q ; q ) n x n . By replacing α by α − µ − , and setting A n = ( − n a µ + n ( q ; q ) µ + n ( q ; q ) n , we, partially, get x α − J (1)2 µ (cid:0) √ ax ; q (cid:1) = x α − X n ≥ A n x n = x α − f ( x ) . Hence, by Theorem 1, we obtain N q (cid:16) x α − J (1)2 µ (cid:0) √ ax ; q (cid:1)(cid:17) ( u ; v ) = (1 − q ) α − u α − v α X n ≥ A n u n v n ( q α ; q ) n Γ q ( α )= (1 − q ) α − u α − v α Γ q ( α ) X n ≥ A n u n v n ( q α ; q ) n . n Some q-Analogues of the Natural Transform and ... Corrollary 6.
Let a be a real number. Then, we have N q (cid:16) J (1)2 µ (2 √ ax ; q ) (cid:17) ( u ; v ) = X n ≥ A n u n ( q α ; q ) n . Further, Corrollary 5 is expressed in terms of q -Sumudu and q -Laplace transforms as in thefollowing results. Corrollary 7 . Let a be a positive real number. Then, the following hold. (i) S q (cid:16) x α − J (1)2 µ (2 √ ax ; q ) (cid:17) ( u ) = (1 − q ) α − u α − Γ q ( α ) X n ≥ A n u n ( q α ; q ) n , (ii) L q (cid:16) x α − J (1)2 µ (2 √ ax ; q ) (cid:17) ( v ) = (1 − q ) α − v α Γ q ( α ) X n ≥ A n v n ( q α ; q ) n , where A n = ( − n a µ + n ( q ; q ) µ + n ( q ; q ) n . Corrollary 6 extends the results to the case of q -Sumudu and q -Laplace transforms as follows. Corrollary 8 . Let a be a real number. Then, the following hold. (i) S q (cid:16) J (1)2 µ (2 √ ax ; q ) (cid:17) ( u ) = P n ≥ A n u n ( q α ; q ) n ; (ii) L q (cid:16) J (1)2 µ (2 √ ax ; q ) (cid:17) ( v ) = 1 v P n ≥ A n ( q α ; q ) n v n . q N Transform of Special Functions
In this section of this paper, we are concerned with the study of the q -analogue q N of somespecial functions. We are precisely concerned with the series representation of the transformand getting some results related to q -Laplace and q -Sumudu transforms. Theorem 9.
Let f ( x ) = P n ≥ A n x n and α > . Then, we have q N (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (cid:16) uv (cid:17) α (1 − q ) α − Γ q ( α ) X n ≥ A n ( q α ; q ) n k (cid:16) uv ; α + n (cid:17) (cid:16) uv (cid:17) n . (37) Proof
Let the hypothesis of the theorem be satisfied for some α > . Then, on account of(22) , we declare that q N (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = 1 (cid:0) − vu ; q (cid:1) ∞ X k ∈ Z q αk X n ≥ A n q kn (cid:16) − vu ; q (cid:17) k = 1 (cid:0) − vu ; q (cid:1) ∞ X n ≥ A n (cid:16) uv (cid:17) α + n X k ∈ Z (cid:18) q kuv (cid:19) α + n (cid:18) − vu ; q (cid:19) k . (38)Hence, by taking into account the fact thatΓ q ( α ) = K ( A ; α )(1 − q ) α − ( − (1 /A ) ; q ) ∞ X k ∈ Z (cid:18) q k A (cid:19) α (cid:18) − A ; q (cid:19) k where K ( A ; α ) = A α − ( − q/α ; q ) ∞ ( − q t /α ; q ) ∞ ( − α ; q ) ∞ ( − αq − t / ; q ) ∞ [25, Equ.(24)] (for A = uv and α =0 S.K.Q. Al-Omari α + n ) , we get q N (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = X k ∈ Z A n (cid:16) uv (cid:17) α + n Γ q ( α + n ) (1 − q ) α + n − K (cid:0) uv ; α + n (cid:1) = (cid:16) uv (cid:17) α (1 − q ) α − Γ q ( α ) X n ≥ A n ( q α ; q ) n K (cid:0) uv ; α + n (cid:1) (cid:16) uv (cid:17) n . This completes the proof of the theorem.As a straightforward corrollary of Theorem 9, the previous theorem (for A = 1 , A n = 0 , for n ≥
1) gives q N (cid:0) x α − (cid:1) ( u ; v ) = (cid:0) uv (cid:1) α − (1 − q ) α − Γ q ( α ) K (cid:0) uv ; α (cid:1) . (39)Also, Equ. (39) reveals : Corrollary 10.
The following hold true. ( i ) q L (cid:0) x α − (cid:1) ( v ) = 1 v α − (1 − q ) α − Γ q ( α ) K (cid:0) v ; α (cid:1) . (40)( ii ) q S (cid:0) x α − (cid:1) ( u ) = u α − (1 − q ) α − Γ q ( α ) K ( u ; α ) . (41)Further, (39) and (41) jointly lead to the conclusion ( q N (1)) ( u ; v ) = 1 K (cid:0) uv ; α (cid:1) . Therefore,we are directed to the results q L (1) ( v ) = 1 K ( v ; 1) and ( q S (1)) ( u ) = 1 K ( u ; 1) . Theorem 11.
Let a be a real number and f ( x ) = m − k Φ m − (cid:20) a , a , ..., a m − k , q α b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q, ax (cid:21) . Then, we have q N (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (cid:0) uv (cid:1) α (1 − q ) α − Γ q ( α ) K (cid:0) uv ; α (cid:1) m − k +1 Φ m − (cid:20) a , a , ..., a m − k , q α b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q, auvq α (cid:21) . Proof
Let the hypothesis of the theorem be satisfied. A charity of (14) gives f ( x ) = ∞ X ( a , a , ..., a m − k ; q ) n ( b , b , ..., b m − , q ) n (cid:16) ( − n q n ( n − (cid:17) k a n ( q ; q ) n x n . On setting A n = ( a , a , ..., a m − k ; q ) n ( b , b , ..., b m − , q ) n (cid:16) ( − n q n ( n − (cid:17) k a n ( q ; q ) n and using Theorem 9 it im-plies q N (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (cid:16) uv (cid:17) α (1 − q ) α − Γ q ( α ) ∞ X A n ( q α ; q ) n K (cid:0) uv ; α + n (cid:1) (cid:16) uv (cid:17) n . n Some q-Analogues of the Natural Transform and ... K ( A, α ) = q α − K ( A, α − , the preceding equation gives q N (cid:0) x α − f ( x ) (cid:1) ( u ; v ) = (cid:16) uv (cid:17) α (1 − q ) α − Γ q ( α ) K (cid:0) uv ; α (cid:1) ∞ X A n ( q α ; q ) n (cid:16) ( − n q n ( n − (cid:17) − (cid:18) − q α (cid:19) n (cid:16) uv (cid:17) n = (cid:0) uv (cid:1) α (1 − q ) α − Γ q ( α ) K (cid:0) uv ; α (cid:1) m − k +1 Φ m − (cid:20) a , a , ..., a m − k , q α b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q, auvq α (cid:21) . Hence the theorem is completely proved.
Corrollary 12.
Let a be a real number and f ( x ) be defined in terms of the q -hypergeometricfunction f ( x ) = m − k +1 Φ m − (cid:20) a , a , ..., a m − k ,b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q ; ax (cid:21) . Then, the following identities hold. ( i ) q L (cid:0) x α − f ( x ) (cid:1) ( v ) = (1 − q ) α − Γ q ( α ) K (cid:0) v ; α (cid:1) m − k +1 Φ m − (cid:20) a , a , ..., a m − k , q α b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q, avq α (cid:21) . ( ii ) q S (cid:0) x α − f ( x ) (cid:1) ( u ) = u α (1 − q ) α − Γ q ( α ) K ( u ; α ) m − k +1 Φ m − (cid:20) a , a , ..., a m − k , q α b , b , ..., b m − (cid:12)(cid:12)(cid:12)(cid:12) q, auq α (cid:21) . Proof is straightforward from Theorem 11. Details are therefore omitted.Let α = m = k = 1 and a > . Then, Corrollary 12 gives q N ( E q ( ax )) ( u ; v ) = uvK (cid:0) uv , (cid:1) Φ (cid:20) q (cid:12)(cid:12)(cid:12)(cid:12) q, − auqv (cid:21) . Hence, it follows ( i ) q L ( E q ( ax )) ( v ) = 1 vK (cid:0) v , (cid:1) Φ (cid:20) q (cid:12)(cid:12)(cid:12)(cid:12) q, − aqv (cid:21) . ( ii ) q S ( E q ( ax )) ( u ) = uK ( u, Φ (cid:20) q (cid:12)(cid:12)(cid:12)(cid:12) q, − auq (cid:21) . Also, readers may easily verify that Φ (cid:20) q (cid:12)(cid:12)(cid:12)(cid:12) q, − auqv (cid:21) = ∞ X (cid:18) auqv (cid:19) n = auqv + au , | au | < qv. (42)From above and the fact that K ( s,
1) = 1 we have the following corrollary.
Corrollary 13.
Let a be a real number. Then, we have q N ( E q ( ax )) ( u, v ) = qu ( qv + au ) , | au | < qv. (43)Hence, (43) indeed reveals( i ) q L ( E q ( ax )) ( v ) = q ( qv + a ) , | a | < qv ; ( ii ) q S ( E q ( ax )) ( u ) = qu ( q + au ) , | au | < q. It may also be mentioned here that Corrollary 13 and the identities q sin x = E q ( ix ) − E q ( − ix )2 i and q cos x = E q ( ix ) − E q ( − ix )2 (44)2 S.K.Q. Al-Omari state, without proof, the following result.
Corrollary 14.
Let a be a real number. Then, we have ( i ) q N ( q sin ax ) ( u ; v ) = − qau q v + a u , | au | < qv. ( ii ) q N ( q cos ax ) ( u ; v ) = q uvq v + a u , | au | < qv. Further, Corrollary 14 suggests to have the following conclusions proclaimed.( i ) q L ( q sin ax ) ( v ) = − qaq v + a , | a | < qv ; ( ii ) q L ( q cos ax ) ( v ) = q vq v + a , | a | < qv. ( iii ) q S ( q sin ax ) ( u ) = − qaq + a u , | au | < qv ; ( iv ) q S ( q cos ax ) ( u ) = q q + a u , | au | < qv. Further results concerning some other special functions can be obtained similarly. q -Analogues of N -Transforms for the q -Fox’s H -Function Let α j and β j be positive integers and 0 ≤ m ≤ N ; 0 ≤ n ≤ M. Due to [27] , the q -analogueof the Fox’s H -function is given as H m,nM,N (cid:20) x ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ( a , α ) , ..., ( a µ , α M )( b , β ) , ( b , β ) , ..., ( b N , β N ) (cid:21) =12 πi Z C m Y j =1 G (cid:0) q b j − β j s (cid:1) n Y j =1 G (cid:0) q − a j + α j s (cid:1) πx sN Y j = m +1 G (cid:0) q − b j + β j s (cid:1) M Y j = n +1 G ( q a j − α j s ) G ( q − s ) sin πs d q s (45)where G is defined in terms of the product G (cid:16) q α (cid:17) = ∞ Y (cid:0) − q α − k (cid:1) − = 1( q α , q ) ∞ . (46)The contour C is parallel to Re ( ws ) = 0 , with indentations in such away all poles of G (cid:0) q b j − β j s (cid:1) , ≤ j ≤ m, are its right and those of G (cid:0) q − a j + α j s (cid:1) , ≤ j ≤ n, are the left of C. The integral converges if Re ( s log x − log sin πs ) < , for large values of | s | on C. Hence, (cid:12)(cid:12) arg ( x ) − w w − log | x | (cid:12)(cid:12) < π, | q | < , log q = − w = − w − iw , where w and w are real numbers.Indeed, for α i = β j = 1 , for all i, j, (45) gives the q -analogue of the Meijer’s G -function G m,nM,N (cid:20) x ; q (cid:12)(cid:12)(cid:12)(cid:12) a , a , ..., a M b , b , ..., b N , (cid:21) = 12 πi Z C m Y j =1 G (cid:0) q b j − s (cid:1) n Y j =1 G (cid:0) q − a j + s (cid:1) πx N Y j = m +1 G ( q − b j + s ) M Y j = n +1 G ( q a j − s ) G ( q − s ) sin πs d q s where 0 ≤ m ≤ N ; 0 ≤ n ≤ M and Re ( s log x − log sin πs ) < n Some q-Analogues of the Natural Transform and ... Theorem 15. ( i ) Let λ be any complex number and k ∈ (0 , ∞ ) . The q -Natural transform N q of the Fox’s H q -Function is given as N q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ( a , α ) , ..., ( a M , α M )( b , β ) , b , ..., ( b N , β N ) (cid:21)(cid:19) ( u ; v ) = u λ v λ +1 G ( q ) H m,n +1 M +1 N (cid:20) λ u k v k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( − λ, k ) , ..., ( a M , α M )( b , β ) , ..., ( b N , β N ) (cid:21) . ( ii ) Let λ be any complex number and k ∈ ( −∞ , . The q -Natural transform N q of theFox’s H q -Function is given as N q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ( a , α ) ..., ( a M , α M )( b , β ) , b , ..., ( b N , β N ) (cid:21)(cid:19) ( u ; v ) = u λ v λ +1 G ( q ) H m +1 ,nM,N +1 (cid:20) λ u k v k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ..., ( a M , α M )(1 + λ, − k ) , ( b , β ) , ..., ( b N , β N ) (cid:21) . Proof
We prove Part ( i ) since proof of Part ( ii ) is similar. By considering (45) , we have N q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ( a , α ) , ..., ( a M , α M )( b , β ) , b , ..., ( b N , β N ) (cid:21)(cid:19) ( u ; v ) =12 πi Z C m Y j =1 G ( q bj − βjz ) n Y j =1 G ( q − aj + αjz ) π ( λ ) zN Y j = m +1 G ( q − bj + βjz ) M Y j = n +1 G ( q aj − αjz ) G ( q − z ) sin πz N q (cid:0) x λ + kz (cid:1) ( u ; v ) d q z. (49)By virtue of Theorem 2, (49) gives N q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ( a , α ) , ..., ( a M , α M )( b , β ) , b , ..., ( b N , β N ) (cid:21)(cid:19) ( u ; v ) =12 πi Z C m Y j =1 G ( q bj − βjz ) n Y j =1 G ( q − aj + αjz ) π ( λ ) zN Y j = m +1 G ( q − bj + βjz ) M Y j = n +1 G ( q aj − αjz ) G ( q − z ) sin πz (1 − q ) λ + kz v λ + kz +1 Γ q ( λ + kz + 1) d q z. The fact that (1 − q ) λ + kz Γ q ( λ + kz + 1) = G ( q ) λ + kz G ( q ) gives N q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ( a , α ) , ..., ( a M , α M )( b , β ) , b , ..., ( b N , β N ) (cid:21)(cid:19) ( u ; v ) =12 πi Z C m Y j =1 G ( q bj − βjz ) n Y j =1 G ( q − aj + αjz ) Y ( λ ) z u λ + kzN Y j = m +1 G ( q − bj + βjz ) M Y j = n +1 G ( q aj − αjz ) G ( q − z ) sin πzv λ + kz +1 G ( q λ + kz ) G ( q ) d q z = 12 πi u λ v λ +1 G ( q ) Z C m Y j =1 G ( q bj − βjz ) n Y j =1 G ( q − aj + αjz ) G ( q λ + kz ) Y (cid:16) λ ukvk (cid:17) zN Y j = m +1 G ( q − bj + βjz ) M Y j = n +1 G ( q aj − αjz ) G ( q − z ) sin πz d q z = u λ v λ +1 G ( q ) H m,n +1 M +1 ,N (cid:20) λ u k v k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( − λ , k ) , ( a , α ) , ..., ( a M , α M )( b , β ) , ..., ( b N , β N ) (cid:21) , for k > i ) of the theorem. Proof of Part ( ii ) is quite similar.Hence the theorem is completely proved. Corrollary 16. ( a ) Let λ be any complex number and k ∈ (0 , ∞ ) . Then, we have S.K.Q. Al-Omari ( i ) L q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ..., ( a M , α M )( b , β ) , ..., ( b N , β N ) (cid:21)(cid:19) ( v ) = 1 v λ +1 G ( q ) × H m,n +1 M +1 ,N (cid:20) λv k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( − λ , k ) , ( a , α ) , ..., ( a µ , α M )( b , β ) , ..., ( b N , β N ) (cid:21) . ( ii ) S q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ..., ( a M , α M )( b , β ) , ..., ( b N , β N ) (cid:21)(cid:19) ( u ) = u λ G ( q ) × H m,n +1 M +1 ,N (cid:20) λ, u k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( − λ , k ) , ( a , α ) , ..., ( a µ , α M )( b , β ) , ..., ( b N , β N ) (cid:21) . ( b ) Let λ be any complex number and k ∈ ( −∞ , . Then, we have ( i ) L q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ..., ( a M , α M )( b , β ) , ..., ( b N , β N ) (cid:21)(cid:19) ( v ) = 1 v λ +1 G ( q ) × H m +1 ,nM,N +1 (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ..., ( a M , α M )(1 + λ, − k ) , ( b , β ) , ..., ( b N , β N ) (cid:21) . ( ii ) S q (cid:18) x λ H m,nM,N (cid:20) λx k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ..., ( a M , α M )( b , β ) , ..., ( b N , β N ) (cid:21)(cid:19) ( u ) = u λ G ( q ) × H m +1 ,nM,N +1 (cid:20) λu k ; q (cid:12)(cid:12)(cid:12)(cid:12) ( a , α ) , ..., ( a M , α M )(1 + λ, − k ) , ( b , β ) , ..., ( b N , β N ) (cid:21) . By virtue of Theorem 15 and the elementary extentions of some q -analogues of sin q x, cos q x, sinh q x and cosh q x in terms of Fox’s H-function; see [26] , we introduce the following exam-ples. Example 1.
Let k = 2 and λ = (1 − q ) in Theorem , then we have N q x (1 − q ) q x ( u ; v ) = √ π (1 − q ) − G ( q ) u (1 − q ) / v (1 − q )24 +1 H , , " (1 − q ) u v ; q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (1 − q ) , (cid:0) , (cid:1) , (0 , , (1 , . Example 2.
On setting k = 2 and λ = (1 − q ) in Theorem , we get N q (cid:18) x (1 − q )24 cos q x (cid:19) ( u ; v ) = √ π (1 − q ) − G ( q ) u (1 − q ) / v (1 − q )24 +1 H , , " (1 − q ) u v ; q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) − (1 − q ) , (cid:17) (0 , , (cid:0) , (cid:1) , (1 , . Example 3.
On setting k = 2 and λ = (1 − q ) , in Theorem we get N q (cid:18) x (1 − q )24 sinh q x (cid:19) ( u ; v ) = √ π (1 − q ) − G ( q ) u iv H , , " − (1 − q ) u v ; q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:16) (1 − q ) , (cid:17)(cid:0) , (cid:1) , (0 , , (1 , . n Some q-Analogues of the Natural Transform and ... Example 4.
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