Mathematics Subject Classification: 33D15, 44A10, 44A20The present paper deals with the evaluation of the q-Laplace transforms
of a product of basic analogues of the Bessel functions. As applications,
several useful special cases have been deduced

Kalla, S.

Purohit, S.

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

ON q-LAPLACE TRANSFORMS OFTHE q-BESSEL FUNCTIONSS.D. Purohit 1 and S.L. Kalla 2AbstractThe present paper deals with the evaluation of the q-Laplace transformsof a product of basic analogues of the Bessel functions. As applications,several useful special cases have been deduced.Mathematics Subject Classification: 33D15, 44A10, 44A20Key Words and Phrases: q-Laplace transforms, basic hypergeometricfunctions, basic analogue of the Bessel function1. IntroductionRecently, Yadav and Purohit [12]-[14] evaluated the q-Laplace images ofa number of q-polynomials and generalized basic hypergeometric functionsof one and more variables, including the basic analogue of Fox’s H-function(due to Saxena, Modi and Kalla [9]), and Purohit, Yadav and Vyas [7]obtained q-Laplace transform of a basic analogue of the I-function (due toSaxena and Kumar [8]).Hahn [6] defined the q-analogues of the well-known classical Laplacetransformφ(s) =∫ ∞0e−stf(t)dt, (1.1)by means of the following q-integrals:qLs {f(t)} = 1(1− q)∫ s−10Eq(qst)f(t)d(t; q), (1.2)190 S.D. Purohit, S.L. KallaandqLs {f(t)} = 1(1− q)∫ ∞0eq(−st)f(t)d(t; q), (1.3)where the q-exponential series (analogues of the classical exponential func-tion) are defined byeq(t) =∞∑n=0tn(q; q)n, (1.4)andEq(t) =∞∑n=0(−1)n qn(n−1)/2tn(q; q)n. (1.5)The basic integration (cf. Gasper and Rahman [4]) is defined by∫ x0f(t)d(t; q) = x(1− q)∞∑k=0qkf(xqk). (1.6)By virtue of (1.6), the operator (1.2) can be expressed asφ(s) ≡ qLs {f(t)} = (q; q)∞s∞∑j=0qjf(s−1qj)(q; q)j. (1.7)The correspondence defined by operators (1.2) and (1.7) shall be denotedsymbolically byf(t) ⊃q φ(s),where the function f(t) is called the original function, and φ(s) is named asthe q-Laplace transform, or q-image of the original function f(t).For real or complex a and |q| < 1, the q-shifted factorial is defined as:(a; q)n ={1 ; if n = 0(1− a)(1− aq) · · · (1− aqn−1) ; if n ∈ N, (1.8)also(x− y)ν = xν∞∏n=0[1− (y/x)qn1− (y/x)qν+n], (1.9)andΓq(a) =(q; q)∞(qa; q)∞(1− q)a−1 =(q; q)a−1(1− q)a−1 , (1.10)where a 6= 0,−1,−2, · · · .ON q-LAPLACE TRANSFORMS OF . . . 191The generalized basic hypergeometric series rΦs(.) (cf. Slater [10]), is:rΦs a1, · · · , ar; q, xb1, · · · , bs; = ∞∑n=0(a1, · · · , ar; q)n xn(q, b1, · · · , bs; q)n , (1.11)where for the convergence of the series (1.11), we require |q| < 1 and |x| < 1if r = s+ 1.The q-analogue of the Bessel function (cf. Gasper and Rahman [4]) isdefined asJν(x; q) =(qν+1; q)∞(q; q)∞(x2)ν2Φ1 0, 0; q,−x2/4qν+1; . (1.12)Abdi [1] has investigated the fundamental properties of the q-Laplacetransforms and established several theorems related to q-images of basicfunctions.The main motivation for this paper is to evaluate the q-Laplace trans-form of product of basic analogues of the Bessel functions. Interesting spe-cial cases of the main result are also discussed.2. q-Laplace image of product of q-Bessel functionsTheorem. Consider a tν−1-weighted product of n different q-Besselfunctions J2µj (2√ajt; q), j = 1, . . . , n. Then, their q-Laplace transform isgiven byqLs{tν−1J2µ1(2√a1t; q)· · ·J2µn(2√ant; q)}=Γq(ν+M)(1−q)ν−M−1aµ11 · · · aµnnsν+MΓq(2µ1+1) · · ·Γq(2µn+1)× Ψ(n)2(qν+M ; q2µ1+1, · · · , q2µn+1; q; −a1s, · · · , −ans), (2.1)where M = µ1 + · · ·+ µn, Re(ν +M) > 0 and Re(s) > 0.P r o o f. To prove (2.1), we putf(t) = tν−1J2µ1(2√a1t; q) · · ·J2µn(2√ant; q)into definition (1.7) and use (1.12), to obtainqLs{tν−1J2µ1(2√a1t; q) · · ·J2µn(2√ant; q)}=(q; q)∞s∞∑j=0qj(q; q)j(qjs)ν−1192 S.D. Purohit, S.L. Kalla× (q2µ1+1; q)∞(q; q)∞(a1qjs)µ1· · · (q2µn+1; q)∞(q; q)∞(anqjs)µn×∞∑m1,··· ,mn=0(−a1qj/s)m1(q2µ1+1; q)m1(q; q)m1· · ·(−anqj/s)mn(q2µn+1; q)mn(q; q)mn. (2.2)Interchanging the order of summations in the right-hand side of the aboveequation (2.2), this yields(q2µ1+1; q)∞(q; q)∞· · · (q2µn+1; q)∞(q; q)∞(q; q)∞aµ11 · · · aµnnsν+µ1+···+µn×∞∑m1,··· ,mn=0(−a1/s)m1(q2µ1+1; q)m1(q; q)m1· · · (−an/s)mn(q2µn+1; q)mn(q; q)mn×∞∑j=0qj(ν+µ1+···+µn+m1+···+mn)(q; q)j.Using equation (1.10), and then summing the inner 0Φ0(.)-series withthe help of the formula0Φ0(−;−; q; t) = 1(t; q)∞ , (2.3)the above expression yields:(q; q)∞(1− q)−2(µ1+···+µn)aµ11 · · · aµnnsν+µ1+···+µnΓq(2µ1 + 1) · · ·Γq(2µn + 1)×∞∑m1,··· ,mn=0(−a1/s)m1(q2µ1+1; q)m1(q; q)m1· · ·× (−an/s)mn(q2µn+1; q)mn(q; q)mn1(qν+µ1+···+µn+m1+···+mn ; q)∞=(q; q)∞(1− q)−2(µ1+···+µn)aµ11 · · · aµnnsν+µ1+···+µnΓq(2µ1 + 1) · · ·Γq(2µn + 1)(qν+µ1+···+µn ; q)∞×∞∑m1,··· ,mn=0(qν+µ1+···+µn ; q)m1+···+mn(q2µ1+1; q)m1 · · · (q2µn+1; q)mn(−a1/s)m1(q; q)m1· · · (−an/s)mn(q; q)mn.Further simplifications lead toΓq(ν +M)(1− q)ν−M−1aµ11 · · · aµnnsν+MΓq(2µ1 + 1) · · ·Γq(2µn + 1)ON q-LAPLACE TRANSFORMS OF . . . 193× Ψ(n)2(qν+M ; q2µ1+1, · · · , q2µn+1; q; −a1s, · · · , −ans), (2.4)whereM = µ1+· · ·+µn and Ψ(n)2 (.) denotes the confluent q-hypergeometricfunction defined asΨ(n)2 (a; c1, · · · , cn; q;x1, · · · , xn)=∞∑m1,··· ,mn=0(a; q)m1+···+mn(c1; q)m1 · · · (cn; q)mnxm11(q; q)m1· · · xmnn(q; q)mn. (2.5)This completes the proof of (2.1).3. Special casesIn this section we evaluate the q-Laplace transforms involving the q-Bessel functions as applications of our main result (2.1). First, we putn = 2 in (2.1) and this yields a q-Laplace image of a product of two q-Besselfunctions, namely:qLs{tν−1J2µ1(2√a1t; q)J2µ2(2√a2t; q)}=Γq(ν +M)(1− q)ν−M−1aµ11 aµ22sν+MΓq(2µ1 + 1)Γq(2µ2 + 1)× Ψ2(qν+M ; q2µ1+1, q2µ2+1; q;−a1s,−a2s), (3.1)where M = µ1 + µ2, Re(ν +M) > 0 and Re(s) > 0.If we put n = 1, µ1 = ν, ν = µ and a1 = a in (2.1), we obtain a Laplaceq-image of basic analogue of the Bessel function as below:qLs{tµ−1J2ν(2√at; q)}=Γq(µ+ ν)(1−q)µ−ν−1aνsµ+νΓq(2ν + 1)1Φ1 qµ+ν ; q,−a/sq2ν+1; ,(3.2)for Re(µ+ ν) > 0 and Re(s) > 0.Replacing µ and ν by ν2 + 1 andν2 respectively in (3.2), yieldsqLs{tν/2Jν(2√at; q)}= aν/2s−ν−1 eq(−a/s), Re(s) > 0. (3.3)Next, we put ν = 1 in (3.3) and getqLs{t1/2J1(2√at; q)}= a1/2s−2 eq(−a/s), Re(s) > 0. (3.4)194 S.D. Purohit, S.L. KallaSimilarly, for ν = 0 the equation (3.3) reduces toqLs{J0(2√at; q)}= s−1 eq(−a/s), Re(s) > 0. (3.5)Again, on replacing µ and ν by 1− ν2 and ν2 respectively, (3.2) yieldsqLs{t−ν/2Jν(2√at; q)}=(1− q)−νaν/2s Γq(ν + 1)1Φ1 q; q,−a/sqν+1; .By further simplifications this reduces toqLs{t−ν/2Jν(2√at; q)}=(−1)ν sν−1aν/2 Γq(ν)eq(−a/s)Γq(ν,−a/s), (3.6)where Γq(α, x) denotes the q-extension of the incomplete gamma functionγ(α, x) (cf. Gupta [5], eqn.(9), p.253):Γq(α, x) =xα(x; q)∞(1− q)1−α(1− qα) 1Φ1 q; q, xqα+1; . (3.7)Further, for ν = 0 and a = 0, we obtain a q-extension of the well-knownresult for the Laplace transform, namely:qLs{tµ−1}=Γq(µ)(1− q)µ−1sµ, Re(s) > 0. (3.8)Finally, it is interesting to observe that in view of the limit formulaelimq→1−Γq(a) = Γ(a) and limq→1−(qa; q)n(1− q)n = (a)n , (3.9)where(a)n = a(a+ 1) · · · (a+ n− 1), (3.10)the main result (2.1) and the results (3.2) to (3.6), give q-extensions of theknown results mentioned in Erde´lyi, Magnus, Oberhettinger and Tricomi[3](table number (4.14), pp. 182-187), namely:L{tν−1J2µ1(2√a1t) · · ·J2µn(2√ant)}=Γ(ν +M)aµ11 · · · aµnnsν+MΓ(2µ1 + 1) · · ·Γ(2µn + 1)× Ψ(n)2(ν +M ; 2µ1 + 1, · · · , 2µn + 1; −a1s, · · · , −ans), (3.11)ON q-LAPLACE TRANSFORMS OF . . . 195where M = µ1 + · · ·+ µn, Re(ν +M) > 0 and Re(s) > 0;L{tµ−1J2ν(2√at)}=Γ(µ+ ν)aνsµ+νΓ(2ν + 1) 1F1 µ+ ν; −a/s2ν + 1; , (3.12)where Re(µ+ ν) > 0 and Re(s) > 0;L{tν/2Jν(2√at)}= aν/2s−ν−1 e−a/s, Re(s) > 0; (3.13)L{t1/2J1(2√at)}= a1/2s−2 e−a/s, Re(s) > 0; (3.14)L{J0(2√at)}= s−1 e−a/s, Re(s) > 0; (3.15)L{t−ν/2Jν(2√at)}=eipiν sν−1aν/2 Γ(ν)e−a/sγ(ν,eipias). (3.16)The results proved in this paper give some contributions to the theoryof the q-series, especially q-Bessel functions, and may find applications tosolutions of certain q-difference and q-integral equations associated withvarious q-Bessel functions. In this regard, one can refer to the work of Abdi[2].References[1] W.A. Abdi, On q-Laplace transforms. Proc. Nat. Acad. Sc. India29(A) (1960), 389-407.[2] W.H. Abdi, Applications of q-Laplace transform to the solution ofcertain q-integral equations. Rend. Circ. Mat. Palermo (2)11 (1962),1-13.[3] A. Erde´lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Tables ofIntegral Transforms. Cambridge University Press, Cambridge (1990).[4] G. Gasper and M. Rahman, Generalized Basic Hypergeometric Series.Cambridge University Press, Cambridge (1990).[5] A. Gupta, A q-extension of incomplete gamma function. J. IndianMath. Soc. 58, No 4 (1992), 249-264.[6] W. Hahn, Beitrage Zur Theorie der Heineschen Reihen, die 24 Inte-grale der hypergeometrischen q-Differenzengleichung, das q-Analogonder Laplace Transformation. Math. Nachr., 2 (1949), 340-379.196 S.D. Purohit, S.L. Kalla[7] S.D. Purohit, R.K. Yadav and V.K. Vyas, On applications of q-Laplacetransforms to a basic analogue of the I-function. (Communicated).[8] R.K. Saxena and R. Kumar, A basic analogue of the generalized H-function. Le Matematiche L, Fasc. II (1995), 263-271.[9] R.K. Saxena, G.C. Modi and S.L. Kalla, A basic analogue of Fox’sH-function. Rev. Tec. Ing. Univ. Zulia, 6 (1983), 139-143.[10] L.J. Slater, Generalized Hypergeometric Functions. Cambridge Uni-versity Press, Cambridge (1966).[11] H.M. Srivastava and P.W. Karlson,Multiple Gaussian HypergeometricSeries. Ellis Harwood Ltd., John Wiley and Sons, New York (1985).[12] R.K. Yadav and S.D. Purohit, On q-Laplace transforms of certaingeneralized basic hypergeometric functions. Proc. 5th Int. Conf.SSFA 5 (2004), 74-81.[13] R.K. Yadav and S.D. Purohit, On q-Laplace transforms of certainmultiple basic hypergeometric functions. The Math. Student 74, No1-4 (2005), 207-215.[14] R.K. Yadav and S.D. Purohit, On q-Laplace transforms of certain q-hypergeometric polynomials. Proc. Nat. Acad. Sci. India 76(A), III(2006), 235-242.1 Department of Basic-Sciences (Mathematics)College of Technology and EngineeringM.P. University of Agriculture and TechnologyUdaipur - 313001, INDIAe-mail: sunil a purohit@yahoo.com Received: September 24, 20072 Department of Mathematics and Computer ScienceFaculty of Science, Kuwait UniversityP.O. Box-5969, Safat 13060, KUWAITe-mail: shyamkalla@yahoo.com

On q-Laplace Transforms of the q-Bessel Functions

http://sci-gems.math.bas.bg/jspui/bitstream/10525/1316/1/fcaa-vol10-num2-2007-189p-196p.pdf

On q-Laplace Transforms of the q-Bessel Functions

Abstract

Similar works

Full text

Available Versions

Bulgarian Digital Mathematics Library at IMI-BAS