The Gauss-Airy functions and their properties

Transcription

1 Annals of the University of Craiova, Mathematics and Computer Science Series Volume 432, 26, Pages 9 27 ISSN: The Gauss-Airy functions and their properties Alireza Ansari Abstract. In this paper, in connection with the generating function of three-variable Hermite polynomials, we introduce the Gauss-Airy function GAix; y, z = e yt2 cosxt + zt3 dt, y, x, z R. 3 Some properties of this function such as behaviors of zeros, orthogonal relations, corresponding inequalities and their integral transforms are investigated. Key words and phrases. Integral transforms. Gauss-Airy function, Airy function, Inequality, Orthogonality,. Introduction We consider the three-variable Hermite polynomials [ n 3 ] 3H n x, y, z = n! j= [ n 3j 2 ] k= as the coefficient set of the generating function e xt+yt2 +zt 3 z j j! y k k!n 2k! xn 3j 2k, e xt+yt2 +zt 3 = 3H n x, y, z tn n!. 2 n= For the first time, these polynomials [8] and their generalization [9] was introduced by Dattoli et al. and later Torre [] used them to describe the behaviors of the Hermite- Gaussian wavefunctions in optics. These are wavefunctions appeared as Gaussian apodized Airy polynomials [3, 7] in elegant and standard forms. Also, similar families to the generating function 2 have been stated in literature, see [, 2]. Now in this paper, it is our motivation to introduce the Gauss-Airy function derived from operational relation 2 d3 2 +z 3H n x, y, z = e y d dx dx 3 x n. 3 For this purpose, in view of the operational calculus of the Mellin transform [4 6], we apply the following integral representation e ys2 +zs 3 = e sξ GAiξ, y, z dξ, 4 Received February 28, 25. 9

2 2 A. ANSARI Figure. The function GAix. where the function GAiξ, y, z is given by GAiξ, y, z = e yr2 cosrξ zr 3 dr. 5 If we set s = d dx in 4 and incorporate it with the relation 3, we see that the threevariable Hermite polynomials is presented by the following integral representation with respect to the Gauss-Airy function as a convolution product of two distributions 3H n x, y, z = ξ n GAiξ x, y, z dξ. 6 The main object of this paper is devoted to the properties of Gauss-Airy function GAi, which are summarized as the follows. In Section 2, we state the Weierstrass infinite product for the entire function GAix with respect to its zeros. We define the Gauss-Airy function zeta and state a formula for obtaining its values. In Section 3, we show that the Gauss-Airy function is logarithmically concave and state some inequality for this function using this property. In Section 4, we get a relationship between the Gauss-Airy function and Airy function, and present orthogonal relations for the Gauss-Airy function with resect to its zeros. In last section, we try to find some identities for the Gauss-Airy function in view of the Mellin, Laplace and Fourier transforms of this function. 2. Zeros of the Gauss-Airy function and the Gauss-Airy zeta function In this section, for describing the behavior of function GAix, y, z, we confine ourself to the function GAix = GAix,, 3. For the other parameters y and z, the same analysis can be applied. As we see in Figure, GAix is an oscillatory function for negative x and tends to zero algebraically. We denote the zeros by, n =, 2,, and show the first six roots of this function, see Table. Also, we

3 THE GAUSS-AIRY FUNCTIONS AND THEIR PROPERTIES 2 ga ga ga ga ga ga Table. The first six real roots of GAix. can write the Weierstrass infinite product for this function as GAix = GAie kx + x e x GAi gan, k = ga n GAi = Now, using the above infinite product and zeros, we define the Gauss-Airy zeta function and evaluate some values for it. Let us start with the following lemma. Lemma 2.. The Gauss-Airy function GAix satisfy the following ordinary differential equation y 2y xy =, x R. 8 Proof. By using the Laplace integral yx = Γ e xr vrdr, 9 and substituting into the ordinary differential equation 8, we get a first order differential equation for vr as follows v r + r 2 2rvr =. After the deformation and normalization of integral 9, we rewrite y as follows y = 2i i i e xr+r2 r3 3 dr, which implies that the solution of differential equation 8 is the Gauss-Airy function. Remark 2.. By the same procedure to Lemma 2., we can show the Gauss-Airy function GAix, a, b satisfy the following differential equation by 2ay xy =, x, b R, a >. 2 Theorem 2.2. The value of the Gauss-Airy zeta function ζ ga s = s, 3 is given by ζ ga s = ds dx s lngaix x=, 4 and in special case, we get ζ ga 2 = k 2 + 2k and ζ ga 3 = 2 kk + k + 2.

4 22 A. ANSARI Proof. If we take the logarithmic derivative of the Weierstrass infinite product, we obtain d dx lngaix = k + x +. 5 Applying the higher order derivatives on the above relation and setting x =, we get the identity 4. The special values ζ ga 2 and ζ ga 3 are given by 4 and differential equation 8 simultaneously. 3. Some inequalities for the Gauss-Airy function In this section, we get some inequalities for the Gauss-Airy function. We start with the logarithmic concavity concept and state the following lemma. Lemma 3.. The Gauss-Airy function GAix is logarithmically concave over ga,, where ga is the first negative root on real axis. Proof. It suffices to show that d2 lngaix dx over the interval ga 2,. For this purpose, by using the Weierstrass infinite product 7 we see that d 2 lngaix dx 2 = + x 2, 6 is negative except at the zeros of GAix. Remark 3.. Since the Gauss-Airy function GAix is logarithmically concave over ga,, then fx = GAix is logarithmically convex over and satisfy the following inequality fux + uy fx u fy u, u, x, y > ga. 7 Theorem 3.2. The following inequalities hold for the Gauss-Airy function GAix xgai 2 x GAi 2 x, x ga,. 2 GAixGAiy GAi 2 x+y 2, x ga,. 3 GAi α xgai α GAiαx, x ga,, α. 4 GAix GAiz, x ga,, z = x + iy. 5 GAiz GAi GAix GAiiy, x, z = x + iy. Proof. To prove these inequalities, we apply the approaches of [4] which generalize the results of the Airy function. For the inequality, we use Lemma 3. for the logarithmic concavity of GAix, which implies that d 2 lngaix dx 2 = xgai2 x GAi 2 x GAi 2 x, 8 and proof is completed. The inequalities 2 and 3 are deduced from Remark 3. by setting u = 2 and u = α, y = in 7, respectively. To prove 4, using the infinite

5 THE GAUSS-AIRY FUNCTIONS AND THEIR PROPERTIES 23 product representation for x ga,, we have GAix GAix + iy = = GAie kx GAie kx+iy + x + x 2 + y2 ga 2 n + x + x+iy e x gan e x+iy gan, 9 which implies that each factor in the infinite product is less than or equal to unit, and proof is completed. Moreover, since for the each factor in the infinite product we have + x + x 2 + y2 ga 2 n = + y 2 ga 2 n + x gan 2 + y2 ga 2 n, 2 and GAiiy = GAi. 2 + y2 ga 2 n Therefore, we can write the following inequality or equivalently GAix GAiz GAi GAiiy, 22 GAiz GAi GAix GAiiy Orthogonality of the Gauss-Airy functions In this section, we intend to introduce an orthonormal basis on the interval, in terms of the Gauss-Airy function and its zeros. Let us start with the following lemma. Lemma 4.. The following relation holds between the Gauss-Airy function and Airy function Aix Ai + x = e x+ 2 3 GAix, 24 where the Airy function is given by [, 2] Aix = cosxr + r3 dr Proof. If we use the differential equation of Gauss-Airy function y 2y xy =, x R, 26 and set zx = Ce x yx, then the above differential equation changes to the following Airy type differential equation z + xz =, x R, 27 with the solution zx = Ai + x. After matching the Gauss-Airy function and Airy function at zero point, we get C = e 2 3 and the relation 24 is obtained.

6 24 A. ANSARI Lemma 4.2. By the same procedure to Lemma 4. and applying differential equation 2, we generalize the relation 24 as follows where a 3 Ai b 2 a a 3 x = e b a x+γ GAix; b, a, 28 γ = ln a 3 Ai b2 a GAi; b, a Remark 4.. The relation 24 implies that the difference between the zeros of Airy function a n and the zeros of Gauss-Airy function is unity, i.e., a n =, n =, 2,. Theorem 4.3. The functions { GAix+gan GAi, n N} form an orthonormal basis on the interval, with respect to the weight function wx = e 2x and zeros. Proof. We recall the following orthonormal basis on the interval, in terms of the Airy function and different zeros a n and a n [5, 6] Aix + a n Aix + a n dx =, n n, n, n N, 3 Ai 2 x + a n dx = Ai 2 a n. 3 At this point, if we substitute the relation 24 into the above integrals, we get an orthonormal basis for the Gauss-Airy function in terms of the zeros and the weight function wx = e 2x e 2x GAix + GAix + dx =, n n, n, n N, 32 e 2x GAi 2 x + dx = GAi Also, in connection with the orthogonality of Airy functions on interval, [6] Ai ξ + a α Aiξ + b β dξ = αβ β 3 α 3 3 δb a, α = β, b a Ai, β α, β 3 α 3 3 we can extend the orthogonality of the Gauss-Airy functions as follows. Theorem 4.4. The functions GAix are orthogonal on the interval, with respect to the weight function of exponential form as follows e ξα+β αβ GAi ξ + a α GAi ξ + b β { δb a, α = β, dξ = 35 α β I, β α, where αβ I = β 3 α 3 3 α + b β 3 b a β 3 α 3 a e 3 GAi b a β 3 α 3 3 β 3 α

7 THE GAUSS-AIRY FUNCTIONS AND THEIR PROPERTIES Integral Transforms of the Gauss-Airy function In this section, we want to obtain the integral transforms type of the Gauss-Airy function. We focus on the Mellin, Laplace and Fourier transforms of this function in view of the Meijer G functions and Airy functions. Theorem 5.. The Mellin transform of the Gauss-Airy function is given by the following relation M{GAix; s} = Γs coss s 2 G,3 3 3,2 4 3, s+ 2, 2,, 2 Γs sins s 2 G,3 3 3,2 4 3, s+ 2, 37,,, where is denoted by k, a = a k, a+ k,, a+k k. Proof. If we use the definition of Mellin transform M{fx; s} = for the Gauss-Airy function GAix GAix = and apply the following facts then, we get the Mellin transform as follows x s fxdx, 38 e r2 cosrx + 3 r3 dr, x s cosrxdx = Γs r s coss, 39 2 x s sinrxdx = Γs r s sins, 4 2 x s GAixdx = Γs coss 2 Γs sins 2 = Γs coss 2 2 Γs sins 2 2 r s e r2 cos 3 r3 dr r s e r2 sin 3 r3 dr r s+ 2 e r cos 3 r 3 2 dr r s+ 2 e r sin 3 r 3 2 dr. At this point, we evaluate the improper integrals in above equation in terms of the Meijer G functions for Rs < 2 as follows [3] page 82 x s GAixdx = Γs coss s 2 G,3 3 3,2 4 3, s+ 2, 2,, 2 Γs sins s 2 G,3 3 3,2 4 3, s+ 2,,,, which confirms the relation 37.

8 26 A. ANSARI In order to obtain the Laplace and Fourier transforms of the Gauss-Airy function, we recall a well-known integral representation for the product of Airy functions as follows [6] AiuAiv = 2 3 Ai2 2 3 ξ 2 + u + v e iu vξ dξ. 4 2 Theorem 5.2. The following Laplace transform type of the Gauss-Airy function holds where the constant γ is given by 29. e ux x GAix; u 2, 4 dx = 4 2e γ Ai 2 u, 42 Proof. Setting u = v in the relation 4, we obtain Ai 2 u = 2 3 Ai2 2 3 ξ 2 + udξ. 43 In this case, after substituting a = 4, b = u 2 into 28 and using the suitable change of variable we easily arrive at 42. Theorem 5.3. The following Fourier transform types hold for the Gauss-Airy function Aiu = where γ is given by 29. e yp2 +izp 3 = e γ 2Ai e ipξ GAiξ, y, z dξ, 44 e 2uξ2 iuξ GAiξ 2, u 2, dξ, 45 4 Proof. To prove 44, it suffice to substitute s = ip into 4. For the relation 45, if we set v = in the relation 4, we obtain Aiu = 2 3 Ai Ai2 2 3 ξ 2 + ue iuξ dξ. 46 Now, by substituting the relation 28 into the above integral with a = 4, b = u 2, we get the relation 45. References [] A. Ansari, M.R. Masomi, Some results involving integral transforms of Airy functions, Integral Transforms and Special Functions , [2] A. Ansari, The fractional Airy transform, Hacettepe Journal of Mathematics and Statistics , [3] A. Ansari, H. Askari, On fractional calculus of A 2n+ x function, Applied Mathematics and Computation , [4] A. Ansari, Some inverse fractional Legendre transforms of gamma function form, Kodai Mathematical Journal , [5] A. Ansari, M. Ahmadi Darani, M. Moradi, On fractional Mittag-Leffler operators, Reports on Mathematical Physics 7 22, 9 3. [6] A. Ansari, A. Refahi, S. Kordrostami, On the generating function e xt+yφt and its fractional calculus, Central European Journal of Physics 23, [7] D. Babusci, G. Dattoli, D. Sacchetti, The Airy transform and associated polynomials, Central European Journal of Physics 96 2,

9 THE GAUSS-AIRY FUNCTIONS AND THEIR PROPERTIES 27 [8] G. Dattoli, S. Lorenzutta, H.M. Svrivastava, A. Torre, Some families of mixed generating functions and generalized polynomials, Le Matematiche , [9] G. Dattoli, B. Germano, P.E. Ricci, Hermite polynomials with more than two variables and associated bi-orthogonal functions, Integral Transforms and Special Functions 2 29, [] A. Torre, Airy polynomials, three-variable Hermite polynomials and the paraxial wave equation, Journal of Optics A: Pure and Applied Optics pp. [] M.A. Khan, A.H. Khan, N. Ahmad, A note on a new three variable analogue of Hermite polynomials of I kind, Thai Journal of Mathematics 92 2, [2] M.A. Khan, G.S. Abukhammash, On Hermite polynomials of two variables suggested by S.F. Ragabs Laguerre polynomials of two variables, Bulletin Calcutta Mathematical Society 9 998, [3] A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and series, direct Laplace transforms vol. 4, Amsterdam, Gordon and Breach, 992. [4] M. Salmassi, Inequalities satisfied by the Airy functions, Journal of Mathematical Analysis and Applications , [5] E.C. Titchmarsh, Eigenfunction expansions, Clarendon Press, Oxford, 962. [6] O. Vallee, M. Soares, Airy functions and applications to physics, Imperial College Press, London, 24. Alireza Ansari - Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahrekord University, P.O. Box 5, Shahrekord, Iran 2- Photonics Research Group, Shahrekord University, P.O. Box 5, Shahrekord, Iran address: alireza 38@yahoo.com