Some results for Laplace-type integral operator in quantum calculus

Transcription

1 Al-Omari et al Advances in Difference Equations 18 18:1 R E S E A R C H Open Access Some results for Laplace-type integral operator in quantum calculus Shrideh KQ Al-Omari 1 DumitruBaleanu 3 and Sunil D Purohit * * Correspondence: sunil_a_purohit@yahoocom Department of HEAS Mathematics Rajasthan Technical University Kota India Full list of author information is available at the end of the article Abstract In the present article we wish to discuss q-analogues of Laplace-type integrals on diverse types of q-special functions involving Fox s H q -functions Some of the discussed functions are the q-bessel functions of the first kind the q-bessel functions of the second kind the q-bessel functions of the third kind and the q-struve functions as well Also we obtain some associated results related to q-analogues of the Laplace-type integral on hyperbolic sine cosine functions and some others of exponential order type as an application to the given theory Keywords: J v x; q function; Y v x; q function; K v x; q function; H v x; q function; Laplace-type integral 1 Introduction and preliminaries Quantum calculus is a version of calculus where derivatives are differences and antiderivatives are sums and no further limits are required The quantum calculus or q-calculus compared to the differential and integral calculus has been very recently named Hence some rules and definitions need to be recalled For < q <1theq-calculusstartswith the definition of the q-analogue of the differential and the q-analogue of derivatives as well The q-analogue of the integer n thefactorialofn and the binomial coefficient are respectively given as [n] q = 1 qn 1 q [n]q!= { n } 1 [k] q n N 1 n = [ ] n = k q n 1 1 q n k+1 1 q k 1 The q-analogue of x + a n n Nanditsq-derivative are respectively given as n 1 x + a n q = x + q j a D q x + a n q =[n] qx + a n 1 q x + a q =1 j= The q-jackson integrals from to a and from a to b are given as follows see [1] see also []: a f x d q x =1 qa f aq k q k 3 The Authors 18 This article is distributed under the terms of the Creative Commons Attribution International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made

2 Al-Omari et al Advances in Difference Equations 18 18:1 Page of 1 and a f x d q x = b f x d q x a b The improper q-jackson integral is given as follows see [1]: f x d q x A f x d q x =1 q n Z q k A f q k A A C The q-analogues of the gamma function are defined by and 1 1 q Ɣ q α= x α 1 E q q1 qx dq x qɣα =KA; α A1 q where α >andforeveryt R x α 1 e q 1 qx dq x Here KA; t=a t 1 q/a; q A; q q t /A; q Aq 1 t ; q n 1 a; q n = 1 aq k a; q = n a; lim q n The very useful identities used in this article are cf [] Ɣ q x= q; q q x ; q 1 q 1 x and a; q t = a; q aq t ; q t R The q-hypergeometric functions are represented by a 1 a a r rφ s α 1 α α s q z = a 1 a a r ; q n z n α 1 α α s ; q n q; q n and m k m 1 a 1 a a m k α 1 α α m 1 q z = a 1 a m k ; q n α 1 α m 1 ; q n [ 1 n q n ] k zn q; q n where a 1 a a p ; q n = p k= a k; q n

3 Al-Omari et al Advances in Difference Equations 18 18:1 Page 3 of 1 H-Function and related functions The H-function which is an extension of the hypergeometric functions p F q introduced by Fox [3]seealso[ 5] has found various applications in a huge range of problems associated with reaction reaction diffusion communication engineering fractional differential equations integral equations theoretical physics and statistical distribution theory as well The H-functions have also been recognized to play a fundamental role in fractional calculus with its applications Fox s H-function admitting to a standard notation is presented as Hpq mn η= 1 jpq mn πi wηw dw 5 P where P is a suitable complex path η w = exp{wlog η + i arg η} jpq mn AwBw w= CsDw and Aw= Cw= m Ɣb j β j w 1 m+1 Bw= n Ɣ1 a j + α j w q p Ɣ1 b j β j w Dw= Ɣa j + α j w 1 n+1 n p 1 m q {a j b j } C {α j β j } R + Letα j and β j be positive integers and m N; n M Then the q-analogue of Fox s H-function is given as see [6] H mn MN x; q a 1 α 1 a α a μ α M b 1 β 1 b β b N β N = 1 m j=1 Gqb j β j s n j=1 Gq1 a j+α j s πx s πi N C j=m+1 Gq1 b j+β j s M j=n+1 Gqa j α j s Gq 1 s sin πs d qs where G is defined in terms of the product G q α = k= 1 q α k 1 = 1 q α ; q 6 The contour C is parallel to Rews = such that all poles of Gq b j β j s 1 j m areits right and those of Gq 1 a j+α j s 1 j naretheleftofc The above integral converges if Res log x log sin πs < for huge values of s on CHence argx w w 1 1 log x < π q <1 log q = w = w1 iw where w 1 and w are real numbers Indeed for α i = β j = 1 for all i j we write the q-analogue of Meijer s G-function as G mn MN a 1 a 1 a M x; q b 1 b b N = 1 m j=1 Gqbj s n j=1 Gq1 aj+s πx πi N C j=m+1 Gq1 bj+s M j=n+1 Gqaj s Gq 1 s sin πs d qs 7 where m N; n M and Res log x log sin πs<

4 Al-Omari et al Advances in Difference Equations 18 18:1 Page of 1 Additionally the q-analogues of the Bessel function J v x of the first kind the Bessel function of Y v x the Bessel function of the third kind K v x and Struve s function H v x are respectively defined in terms of Fox s H q -function by [7] as follows: J v x; q= { Ga } H 1 3 Y v x; q= { Ga } H 1 K v x; q=1 qh 3 1 q H v x; q= H 31 x 1 q 1 ; q x 1 q ; q v 1 v x 1 q v 1 ; q 1 v 1 v v x 1 q ; q v 1 v α 1+α 1 v 1 v 1v+α In [8] seealso[9] some q-analogues of the natural exponential functions sine functions cosine functions hyperbolic sine functions and hyperbolic cosine functions are respectively given in terms of Fox s H-function as follows: e q x=gqh 1 x1 q; q sin q x= π1 q 1 { } Gq x H 1 1 q 3 ; q cos q x= π1 q 1 { } Gq H 1 x 1 q 3 ; q π sinh q x= 1 q 1 { } Gq i H 1 3 x 1 q ; q cosh q x= π1 q 1 { } Gq H 1 3 x 1 q 1 ; q Ontheotherhandsomeimpressiveintegraltransformsalsohavethecorrespondingqanaloguesin the conceptof q-calculus; they include the q-laplace transforms [1] the q- Sumudu transforms [ ] the q-wavelet transform [1] the q-mellin transform [15] q-e 1 -transform [16] q-mangontarum transforms [17 18] q-natural transforms [19] and

5 Al-Omari et al Advances in Difference Equations 18 18:1 Page 5 of 1 so on Recently a number of authors have studied various image formulas for these q- integral transforms associated with a variety of special functions In this sequel we aim to investigate the q-analoguesof Laplace-typeintegrals on diverse types of q-special functions involving Fox s H q -function 3 q-laplace-type transforms for H q -function A Laplace-type integral was introduced in [ 1] The q-analogues of the Laplace-type integral of the first kind were defined later by [] as follows: ql f ξ; y = 1 1 q y 1 = q ; q [] q y i= ξe q q y ξ f ξ dξ q i q ; q i f q i y 1 17 whereas the q-analogues of the Laplace-type integral of the second kind were defined by 1 ql f ξ; y = ξe 1 q q y ξ d q ξ 1 = q i f q i y ; q [] q y ; q i 18 i Z For the sake of convenience we establish some formulas for the q L operator A similar argument can give certain corresponding results for the operator q l Theorem 1 Let β be a positive real number Then ql ξ β y= q ; q [] q y q β ; q Proof By using 17 we have ql ξ β ; y = q ; q [] q y β = q ; q [] q y β i= i= q i q ; q i q i y 1 β q βi y β q ; q i That is ql ξ β ; y = q ; q [] q y i= q βi q ; q i 19 By the fact that e q z= i= z i q; q i

6 Al-Omari et al Advances in Difference Equations 18 18:1 Page 6 of 1 we have ql ξ β ; y = q ; q e [] q y q q β = q ; q [] q y 1 q β ; q This completes the establishment of the belief Theorem Let λ be a complex number Then ql x λ H mn MN γ x k ; q a 1 α 1 a M α M y b 1 β 1 b N β N = q ; q H mn+1 γ λ k a 1 α 1 a M α M y λ+ M+1N q [] q yk b 1 β 1 b N β N where n mand m Nandλ is an arbitrary complex number Proof Let λ be a complex number Then by 17weobtain ql x λ H mn MN γ x k ; q a 1 α 1 a M α M y b 1 β 1 b N β N = 1 m j=1 Gqb j β j z n j=1 Gq a j+α j z πγ z πi N c j=m+1 Gq b j+β j z M j=n+1 Gqa j α j z Gq z sin πz q L x λ+kz y d q z Let β = λ + kz + 1 then by Theorem 1 we have ql x λ+kz y= q L x B y= By invoking 1in we get q ; q [] q y q λ+zk+1 ; q 1 ql x λ H mn MN γ x k ; q a 1 α 1 a M α M y b 1 β 1 b N β N = 1 m j=1 Gqb j β j z n j=1 Gq a j+α j z πγ z πi N c j=m+1 Gq b j+β j z M j=n+1 Gqa j α j z Gq z sin πz By inserting the identity q ; q [] q y q λ+zk+1 ; q d q z G q λ+kz+ 1 = q λ+kz+ ; q

7 Al-Omari et al Advances in Difference Equations 18 18:1 Page 7 of 1 in yields ql x λ H mn MN γ x k ; q a 1 α 1 a M α M y b 1 β 1 b N β N = q ; q m j=1 Gqb j β j z n j=1 Gq a j+α j z πiy λ+ [] N q c j=m+1 Gq b j+β j z M j=n+1 Gqa j α j z Gq1+λ+kz γ z Gq 1 z sin πz π d y q z k Now on account of the definition of H q -function we may establish that ql x λ H mn MN γ x k ; q a 1 α 1 a M α M y b 1 β 1 b N β N = q ; q y λ+ [] q H n+1m NM+1 γ x k ; q 1 b 1 β 1 1 b N β N 1 + λ k 1 a α 1 1 a M α M provided k < The proof is completed Applications to trigonometric and hyperbolic functions In this part we shall give certain natural relevance to the leading results Theorem 3 Let e q be defined in terms of 1 Then ql eq x y= Gq q ; q H 11 1 q [] q y 1 ; q 1 y Proof By setting λ =γ =1 q andk =1Theorem3 immediately follows from Theorem The demonstration of this theorem is finished Theorem Let sin q be defined in terms of 13 Then we have ql sinq x y= π1 q 1 {Gq } H q ; q [] q y 1 q ; q 1 y Proof The proof of this theorem indeed follows from substituting the values λ =k =1 and γ = 1 q and from multiplying by π1 q 1 {Gq } Hence the proof is completed

8 Al-Omari et al Advances in Difference Equations 18 18:1 Page 8 of 1 Theorem 5 Let cos q be defined in terms of 1 Then ql cosq x y= π1 q 1 {Gq } H q ; q [] q y 1 q ; q 1 y Proof Proof follows from Theorem for λ =k =1γ = 1 q The proof is completed Theorem 6 Let sinh q be defined in terms of 15 Then ql sinhq x π1 q 1 {Gq } y= q ; q i[] q y H 11 1 q 13 ; q 1 y Proof By using the special case λ =k =1γ = 1 q The proof is completed Theorem 7 Let cosh q be defined in terms of 16 Then ql coshq x y= π1 q 1 {Gq } H q ; q [] q y 1 q ; q 1 y Proof The validation of this theorem is identical to that of the previous theorem Theorem 8 Let the Bessel function be defined in terms of 8 Then ql Jv x; q y= {Gq } q ; q [] q y H 11 1 q 13 ; q 1 y v 1 v Proof By setting λ =k =1γ = 1 q and multiplying by {Gq } the result follows Theorem 9 Let the q-bessel function of the second kind be defined in terms of 9 11 Then ql Yv x; q y= {Gq } q ; q [] q y H 1 1 q ; q y 1 v 1 v 1 v 1 v 1 111

9 Al-Omari et al Advances in Difference Equations 18 18:1 Page 9 of 1 ql Kv x; q y= 1 q q ; q [] q y H 1 1 q 13 ; q 1 y v 1 v ql Hv x; q y= 1 q 1 α q ; q 1 α [] q y H 3 1 q ; q 1 1 α y 1 v 1 v 11+α 111 Proof Proof of this theoremfollowsfrom 9 11 and the technique quite similar to that of Theorems 3 8 We omit the details Acknowledgements The authors are thankful to the referee for his/her valuable remarks and comments for the improvement of the paper Competing interests The authors declare that they have no competing interests Authors contributions The authors contributed equally and significantly in writing this paper All authors read and approved the final manuscript Author details 1 Department of Basic Sciences Faculty of Engineering Technology Al-Balqa Applied University Amman Jordan Department of Mathematics Cankaya University Ankara Turkey 3 Institute of Space Sciences Magurele-Bucharest Romania Department of HEAS Mathematics Rajasthan Technical University Kota India Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Received: 13 February 18 Accepted: 16 March 18 References 1 Kac VG Cheung P: Quantum Calculus: Universitext Springer New York Gasper G Rahman M: Generalized Basic Hypergeometric Series Cambridge University Press Cambridge Fox C: The G and H-functions as symmetrical Fourier kernels Trans Am Math Soc Mathai AM Saxena RK: Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences Springer Berlin Mathai AM Saxena RK: The H-FunctionwithApplicationinStatisticsandOtherDisciplines Wiley NewYork Saxena RK Modi GC Kalla SL: A basic analogue of Fox s H-function Rev Téc Fac Ing Univ Zulia Saxena RK Kumar R: Recurrence relations for the basic analogue of the H-function J Nat Acad Math Yadav RK Purohit SD Kalla SL: On generalized Weyl fractional q-integral operator involving generalized basic hypergeometric functions Fract Calc Appl Anal Albayrak D Purohit SD Ucar F: On q-sumudu transforms of certain q-polynomials Filomat Abdi WH: On q-laplace transforms Proc Natl Acad Sci India Albayrak D Purohit SD Ucar F: On q-analogues of Sumudu transform An Ştiinţ Univ Ovidius Constanţa Ser Mat Albayrak D Purohit SD Ucar F: Certain inversion and representation formulas for q-sumudu transforms Hacet J Math Stat Purohit SD Ucar F: An application of q-sumudu transform for fractional q-kinetic equation Turk J Math Fitouhi A Bettaibi N: Wavelet transforms in quantum calculus J Nonlinear Math Phys Fitouhi A Bettaibi N: Applications of the Mellin transform in quantum calculus J Math Anal Appl SalemAUcarF:Theq-analogue of the E 1 -transform and its applications Turk J Math Mangontarum MM: On a q-analogue of the Elzaki transform called Mangontarum q-transform Discrete Dyn Nat Soc 1 Article ID Al-Omari SKQ: On q-analogues of the Mangontarum transform for certain q-bessel functions and some application J King Saud Univ Sci Al-OmariSKQ:On q-analogues of the Natural transform of certain q-bessel functions and some application Filomat

10 Al-Omari et al Advances in Difference Equations 18 18:1 Page 1 of 1 Yürekli O: Theorems on L -transforms and its applications Complex Var Elliptic Equ Yürekli O: New identities involving the Laplace and the L -transforms and their applications Appl Math Comput Ucar F Albayrak D: On q-laplace type integral operators and their applications J Differ Equ Appl