In this paper, using the q^2 -Laplace transform early introduced by Abdi [1], we study q-Wave polynomials related with the q-difference operator ∆q,x . We show in particular that they are linked to the q-little Jacobi polynomials p_n (x; &alpha;, &beta; | q^2 )

Fitouhi, Ahmed

Nemri, Akram

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXV (2010) – Fasc. I, pp. 73–82doi: 10.4418/2010.65.1.5POLYNOMIALS EXPANSIONS FOR SOLUTION OFWAVE EQUATION IN QUANTUM CALCULUSAKRAM NEMRI - AHMED FITOUHIIn this paper, using the q2-Laplace transform early introduced by Abdi[1], we study q-Wave polynomials related with the q-difference operator∆q,x. We show in particular that they are linked to the q-little Jacobi poly-nomials pn(x;α,β | q2).1. Introduction and preliminariesIn a recent paper [6], the authors have shown that the solutions of certain q-elliptic problem can be expressed in terms of solutions of a parabolic problemby means of the inverse q2-Laplace transform.In this paper, our interest is to obtain series representations of solutions ofa q-Wave problem. The initial data in these cases is taken to be analytic, andthe representations sets of polynomials involve the q-Laguerre polynomials andq-little Jacobi polynomials. These polynomials are obtainable from the q-Heatpolynomials studied by A. Fitouhi and F. Bouzeffour [3] by the use of the inverseq2-Laplace transform. We also study the series representations of solutions ofthe q-Wave problem concerning the q-difference operator ∆q,x.Throughout this paper, we fix q∈]0,1[ and suppose that log(1−q2)/logq2 ∈N. We recall some usual notions and notations used in the q-theory.Entrato in redazione: 9 novembre 2009AMS 2000 Subject Classification: 33D60, 26D15, 33D05, 33D15, 33D90.Keywords: Quantum calculus, Wave polynomial, q-analysis, q-Integral Transform.74 AKRAM NEMRI - AHMED FITOUHIThe q-shifted factorials are defined by(a;q)0 = 1, (a;q)n =n−1∏k=0(1−aqk), (a;q)∞ =∞∏k=0(1−aqk) (1)and more generally:(a1, · · · ,ar;q)n =r∏k=1(ak;q)n. (2)A basic hypergeometric series isrϕs(a1, · · · ,ar;b1, · · · ,bs;q,x) =∞∑n=0(a1, · · · ,ar;q)n(b1, · · · ,bs;q)n(q;q)n [(−1)nqn(n−1)2 ]1+s−rxn.A function f is said to be q-regular at zero [2] if limn→∞ f (xqn) = f (0) existsand does not depend of x. The q-derivative Dq f [9] of a function f is definedby:Dq,x f (x) =f (x)− f (qx)(1−q)x , x 6= 0. (3)The q-derivative at zero [2] is defined byDq,x f (0) = limn→+∞f (xqn)− f (0)xqn, (4)where the limit exists and independent of x.For n ∈ N,Dnq,x f (x) =(−1)nxn(1−q)nn∑k=0(−1)k (q;q)n(q;q)n−k(q;q)kq−(n−k)(n−k−1)/2 f (qn−kx) (5)The q-analogue of (a+b)n is a non commutative term (a+b)nq given by(a+b)nq ={an(−ba ;q)n, a 6= 0qn(n−1)/2bn, a = 0.(6)It is clear that (a+b)nq and (b+a)nq are not always the same.Some q-functional spaces will be used to establish our result. We begin byputtingRq = {±qk,k ∈ Z}∪{0}, Rq,+ = {+qk,k ∈ Z} (7)and we define Eq,∗(Rq) the space of even functions infinitely q-differentiable atzero.POLYNOMIALS EXPANSIONS FOR SOLUTION OF WAVE EQUATION ... 75We also denote[x]q =1−qx1−q , [n]q! =(q;q)n(1−q)n . (8)The q-shift operators are(Λq,x f )(x) = f (qx), (Λ−1q,x f )(x) = Λq−1,x f (x). (9)We consider the q-difference operator∆q,x = Λ−1q,xD2q,x. (10)Koornwinder and Swarttouw introduced q-trigonometric function denoted in[10] by cos(x;q2) and sin(x;q2), we have in particular:cos(x;q2) = 1ϕ1(0,q,q2;(1−q)2x2) =∞∑n=0(−1)nbn(x;q2) (11)where we have putbn(x;q2) = bn(1;q2)x2n = qn(n−1)(1−q)2n(q;q)2nx2n. (12)More generally the normalized q-Bessel function [4] is given byjα(x;q2) = Γq2(α+1)qn(n−1)qα(1+q)αxαJα((1−q)x;q2) (13)=∞∑n=0(−1)nbn,α(x,q2) (14)where Jα(x;q2) is the Hahn Exton q-Bessel function [12] andbn,α(x,q2) = bn,α(1,q2)x2n =Γq2(α+1)qn(n−1)(1+q)2nΓq2(n+1)Γq2(α+n+1)x2n. (15)The q- jα Bessel function jα(x;q2) is entire function and tends to the normalizedjα Bessel function as q−→ 1−.One can see, after simple computation, thatj− 12 (x;q2) = cos(x;q2), (16)j 12(x;q2) =sin(x;q2)x. (17)76 AKRAM NEMRI - AHMED FITOUHIThe q2-Jackson integral from 0 to a and from 0 to ∞ are respectively defined by∫ a0f (x)dq2x = (1−q2)a∞∑n=0f (aq2n)q2n,∫ ∞0f (x)dq2x = (1−q2)+∞∑−∞f (q2n)q2n.Note that for n ∈ Z and a ∈ Rq, we have∫ ∞0f (q2nx)dq2x =1q2n∫ ∞0f (x)dq2x,∫ a0f (q2nx)dq2x =1q2n∫ aq2n0f (x)dq2x.(18)The q2-integration by parts is given for suitable function f and g regular at zeroby: ∫ baf (x)Dq2,xg(x)dq2x =[f (x)g(x)]ba−∫ baf (q2x)Dq2,xg(x)dq2x. (19)The improper integral is defined in the following way (see [10]; [11])∫ ∞/A0f (x)dq2x = (1−q2)+∞∑−∞f(q2nA)q2nA, A 6= 0. (20)We remark that, for n ∈ Z, we have∫ ∞/q2n0f (x)dq2x =∫ ∞0f (x)dq2x. (21)The q2-analogue of the exponential function [7] are given byEq2(x) = 0ϕ0(−,−;q2,−(1−q2)x) =∞∑n=0qn(n−1)(1−q2)n(q2;q2)nxn (22)= (−(1−q2)x;q2)∞, for x ∈ Candeq2(x) = 1ϕ0(0,−;q2,(1−q2)x) =∞∑n=0(1−q2)n(q2;q2)nxn (23)=1((1−q2)x;q2)∞ , for |x|<11−q2 .They satisfyeq2(x).Eq2(−x) = 1, Dq2,xeq2(x) = eq2(x), Dq2,xEq2(x) = Eq2(q2x). (24)POLYNOMIALS EXPANSIONS FOR SOLUTION OF WAVE EQUATION ... 77The little q-Jacobi polynomials [7] is defined bypn(x;α,β | q) = q(n+α)n(q−n−α ;q)n(qn+α+β+1;q)n2ϕ1(q−n,qn+α+β+1;qα+1;q,qx) (25)=q(n+α)n(q−n−α ;q)n(qn+α+β+1;q)nn∑k=0(q−n;q)k(qn+α+β+1;q)k(qα+1;q)kqkxk(q;q)k.Jackson [8] defined a q2-analogue of the Gamma function byΓq2(x) =(q2;q2)∞(q2x;q2)∞(1−q2)1−x. (26)Abdi in [1], introduced the q2-Laplace transform byϕ(s) = q2Ls{ f (t)}t→s (27)=∫ 1/(1−q2)s0Eq2(−q2st) f (t)dq2t (28)=∫ ∞/(1−q2)s0Eq2(−q2st) f (t)dq2t. (29)Moreover, since log(1−q2)/ log(q2)∈N and s∈Rq,+, we obtain from (21)that the following q2-integral representations hold:ϕ(s) = q2Ls{ f (t)}t→s =∫ ∞/s0Eq2(−q2st) f (t)dq2t =∫ ∞0Eq2(−q2st) f (t)dq2t(30)and (see [5] Theorem 1 )Γq2(s) =∫ +∞0ts−1Eq2(−q2t)dq2t. (31)In [6], we have defined the q-Wave polynomials associated with the q-difference operator ∆q,x by defining w1,2n and w2,2n given, for x, t in R, by:w1,2n(x, t;q2) = [2n]q!n∑k=0qk2bn−k(x;q2)t2k[2k]q!w2,2n(x, t;q2) = [2n]q!n∑k=0qk2bn−k(x;q2)t2k+1[2k+1]q!.(32)wich can be expressed in term of q-little Jacobi polynomial pn(x;α,β | q2) [6]as follows78 AKRAM NEMRI - AHMED FITOUHIProposition 1.1. For x, t in Rq we have:w1,2n(x, t;q2) = (−1)nq−n(n−1)(q2;q2)nt2n pn(q2nx2/q2t2; -1/2, -2n | q2)w2,2n(x, t;q2) = (−1)nq−n(n+1)(q2;q2)nt2n+1 pn(q2nx2/t2; -1/2, -2n-1 | q2).Proposition 1.2. For x, t in R we have:| w1,2n(x, t;q2) | ≤ q−n2 [2n]q!2 [(x+qnt)2nq +(x−qnt)2nq ]| w2,2n(x, t;q2) | ≤ q−n2 [2n]q!2 t[(x+qnt)2nq +(x−qnt)2nq ]where (a+b)nq is given by (6).Proof. Owing to the relation(q−2n;q2)k = (−1)kqk(k−1)−2nk (q2;q2)n(q2;q2)n−k,we deduce, from Proposition 1.1, thatw1,2n(x, t;q2) == (−1)nqn2(q;q)2ntn∑k=0(−1)k qk(k−1)qk(q−2n;q2)k(q;q2)k(q2;q2)k(q2;q2)n−k(q;q)2n−2kx2kt2n−2k−1= (−1)nqn2(q;q)2nt2nn∑k=0qk(k−1)qk2−2nk(q2;q2)n(q;q2)k(q2;q2)k(q2;q2)n−k(q;q2)n−kx2kt2k= (−1)nqn2(q;q)2nt2nn∑k=0[nk]q2qk(k−1)qk2−2nk(q;q2)k(q;q2)n−kx2kt2k.Hence| w1,2n(x, t;q2) | ≤ qn2 (q;q)2n(1−q)n t2nn∑k=0[nk]q2qk(k−1)[x2q2nt2]k≤ qn2 (q;q)2n(1−q)n t2n(1+x2q2nt2)nq≤ q−n2 [2n]q!(x2+q2nt2)nq.The result is then deduced by the fact that:(x2+q2nt2)nq =12[(x+qnt)2nq − (x−qnt)2nq]. (33)POLYNOMIALS EXPANSIONS FOR SOLUTION OF WAVE EQUATION ... 79Proposition 1.3. For x, t in R and n≥ 0, we have:|Dq,tw1,2n(x, t;q2)| ≤ q4nC(q,n)t[(x+qnt)2n−2q +(x−qnt)2n−2q]|∆q,tw1,2n(x, t;q2)| ≤C(q,n)(x2+ t2+q(1+q)[n−1]q2t2)[(x+qnt)2n−4q +(x−qnt)2n−4q].Furthermore|∆q,xw1,2n(x, t;q2)| ≤C(q,n)(x2+q2(n−1)t2+q(1+q)×[n−1]q2x2)[(x+qn−1t)2n−4q +(x−qn−1t)2n−4q ]where the constant C(q,n) is given byC(q,n) = q−n2−2n 1+q2[2n]q! [n]q2 .2. Convergence of series+∞∑n=0αn[2n]q!w1,2n(x, t;q2) and+∞∑n=0αn[2n]q!w2,2n(x, t;q2)In this section, we prove that the series+∞∑n=0αn[2n]q!w1,2n(x, t;q2) and+∞∑n=0αn[2n]q!w2,2n(x, t;q2) converge in the strip{(x, t)/ | x |+ | t |< R}, R > 0.Given R0 such that the previous series converge for | x |< R0. We considerthe q-difference problems (I) and (II) given by:(I)∆q,tw(x, t;q2) = ∆q,xw(x, t;q2)w(x,0;q2) = φ(x)Dq,tw(x, t,q2)|t=0 = 0(II)∆q,tw(x, t;q2) = ∆q,xw(x, t;q2)w(x,0;q2) = 0Dq,tw(x, t,q2)|t=0 = φ(x)where ∆q,. is given by (10) and φ being an entire even function defined on C,infinitely q-differentiable at zero, having the following expansion:φ(x) =+∞∑n=0αnbn(x;q2) (34)the convergence holds for | x |< R0.In this section, We prove the solutions of (I) and (II) have respectively an ex-pansion of the form+∞∑n=0αn[2n]q!w1,2n(x, t;q2) and+∞∑n=0αn[2n]q!w2,2n(x, t;q2), whichconverge respectively in the strip{(x, t)/ | x |+ | t |< R}, R > 0.80 AKRAM NEMRI - AHMED FITOUHITheorem 2.1. Let (αn)n be a sequence of real or complex numbers such thatthe series ∑αnbn(x;q2) converge for any sequence of real or complex numbers,for all | x |< R0. Then:i) the series+∞∑n=0αn[2n]q!w1,2n(x, t;q2) (35)is solution of the q-problem (I) in the strip{(x, t)/ | x | + | t |< R0}andconverges uniformly in any compact subset of this strip.ii) the series+∞∑n=0αn[2n]q!w2,2n(x, t;q2) (36)is the solution of the q-problem (II) in the strip{(x, t)/ | x | + | t |< R0}and converges uniformly in any compact subset of this strip.Proof. Given R1 < R0 and K ={(x, t)/ | x | + | t |< R1}, then for all (x, t) inK, | x+qnt |< R1 and | x−qnt |< R1. Furthermore, the fact that+∞∑n=0αnbn(x;q2)converges for | x |< R0 implies that there exist M > 0 such that| αn |≤ Mq−n2 [2n]q!R2n1. (37)The Proposition 1.2, give|+∞∑n=0αn[2n]q!w1,2n(x, t;q2)| ≤ M2+∞∑n=01R2n1[(x+qnt)2nq +(x−qnt)2nq ].Hence, the convergence of the last series holds for all (x, t) in K then+∞∑n=0αn[2n]q!w1,2n(x, t;q2) converges uniformly in any compact subset of K tow(x, t;q2) and we havelimt→0w(x, t) = φ(x). (38)Now using Proposition 1.3, we can deduce easily that+∞∑n=0αn[2n]q!Dq,tw1,2n(x, t;q2)converges uniformly in any compact subset of K with Dq,tw(x, t;q2) as sum andlimt→0Dq,tw(x, t;q2) = 0. (39)So i) is then proved.To prove ii), we proceed with the same way.POLYNOMIALS EXPANSIONS FOR SOLUTION OF WAVE EQUATION ... 81Acknowledgment. The authors would like to thank the Board of Editors fortheir helpful comments. They are thankful to the anonymous reviewer for thehelpful remarks, valuable comments and suggestions.REFERENCES[1] W. H. Abdi, On q-Laplace transforms, Proc. Nat. Acad. Sc (India) 29 (1960), 389–408.[2] M. H. Annaby - Z. S. Mansour, Basic Sturm-Liouville problems, J. Phys. A 38 (17)(2005), 3775–3797.[3] F. Bouzeffour, q-cosine Fourier Transform and q-Heat Equation, These d’etat(2009).[4] A. Fitouhi - M. Hamza - F. Bouzeffour, The q-Jα Bessel function, J. Approx. Theory115 (2002), 114–116.[5] A. Fitouhi - N. Bettaibi - K. Brahim, The Mellin Transform in Quantum Caculus,Constr. Approx 23 (2006), 305–323.[6] A. Fitouhi - A. Nemri - W. Binous, On the connection between q-heat and q-waveproblems, to appear.[7] G. Gasper - M. Rahman, Basic Hypergeometric series, Encyclopedia of mathemat-ics and its applications 35, Cambridge University Press, 1990.[8] F. H. Jackson, On q-Definite integrals, Quart. J. Pure. Appl. Math 41 (1910), 193–203.[9] F. H. Jackson, On q-Function and certain Difference Operator, Trans. Royal. Soc.London 46 (1908), 253–281.[10] T. H. Koornwinder - R.F. Swarttouw, On q-Analogues of the Fourier and Hankeltransforms, Trans. Amer. Math. Soc. 333 (1992), 445–461.[11] V. G. Kac - P. Cheeung, Quantum calculus, Universitext, Springer-Verlag, NewYork, 2002.[12] R. F. Swarttouw, The Hahn–Exton q-Bessel Function, Ph. D. Thesis, Delft Tech-nical University, 1992.82 AKRAM NEMRI - AHMED FITOUHIAKRAM NEMRIDe´partement de MathematiquesFaculte´ des Sciences de Tunis1060 Tunis, Tunisia.e-mail: Akram.Nemri@fst.rnu.tnAHMED FITOUHIDe´partement de MathematiquesFaculte´ des Sciences de Tunis1060 Tunis, Tunisia.e-mail: Ahmed.Fitouhi@fst.rnu.tn

Polynomial expansions for solution of wave equation in quantum calculus

In this paper, using the q^2 -Laplace transform early introduced by Abdi [1], we study q-Wave polynomials related with the q-difference operator ∆q,x . We show in particular that they are linked to the q-little Jacobi polynomials p_n (x; &alpha;, &beta; | q^2 ).<br /

Akram Nemri

Ahmed Fitouhi

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Polynomial expansions for solution of wave equation in quantum calculus

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