We employ an umbral formalism to reformulate the theory of Hermite polynomials and the derivation of the associated lacunary generating functions

D. Babusci

G. Dattoli

GERMANO, Bruna

MARTINELLI, Maria Renata

P. E. Ricci

English

Archivio della ricerca- Università di Roma La Sapienza

Ilirias Journal of MathematicsISSN: 2334-6574, URL: http://www.ilirias.comVolume 4 Issue 1(2015), Pages 16-23.LACUNARY GENERATING FUNCTIONS OF HERMITEPOLYNOMIALS AND SYMBOLIC METHODSG. DATTOLI, B. GERMANO, M.R. MARTINELLI, P.E. RICCIAbstract. We employ an umbral formalism to reformulate the theory of Her-mite polynomials and the derivation of the associated lacunary generatingfunctions.1. IntroductionIn a paper of few years ago Gessel and Jayawant [6] have discussed a triplelacunary generating function for Hermite polynomials. The Authors employ twodifferent methods, one of umbral nature, the other based on combinatorial argu-ments.In this paper we comment on the umbral technique proposed in [6], discussits link with previously developed formalisms and suggest extensions, allowing theumbral treatment of families of Hermite like polynomials.The 2-variable polynomials, defined by the series [1]Hn(x, y) = n![n2 ]∑r=0xn−2 ryr(n− 2 r)! r! (1.1a)belong to the Hermite-like polynomials. This family of polynomials has manygeneralization and sometimes there is some confusion in the literature, regarding therelevant notation. For reasons which will be clarified in the following, they shouldbe denoted by H(2)n (x, y) and should be referred to as “second order two variableHermite polynomials”, we will however keep the upper index only for polynomialswith order > 3 or add it whenever strictly necessary to avoid confusion.The function generating (1.1a) reads∞∑n=0tnn!Hn(x, y) = ex t+y t2 . (1.1b)A remarkable property is the operational definition [3]Hn(x, y) = ey ∂2xxn (1.2)2000 Mathematics Subject Classification. 33C45, 33F10.Key words and phrases. Hermite polynomials, Lacunary generating functions, Symbolic andumbral methods.c©2015 Ilirias Publications, Prishtine¨, Kosove¨.Submitted May 29, 2015. Published Jul 21, 2015.16LACUNARY GENERATING FUNCTIONS OF HERMITE POLYNOMIALS 17due to the fact that they are a solution of the partial differential equation∂yF (x, y) = ∂2xF (x, y), (1.3)with the “initial condition”F (x, 0) = xn . (1.4)For this reason they are also defined “heat polynomials” [10].From equation (1.1a) we derive the boundary condition at x = 0Hn(0, y) = n!yn2Γ(n2 + 1) ∣∣∣cos(n pi2)∣∣∣ . (1.5)The use of equation (1.2) is extremely useful, for example we can derive straight-forwardly the double lacunary Hermite generating function, which can be formallywritten as∞∑n=0tnn!H2n(x, y) = ey ∂2xex2t . (1.6)A definite meaning to the rhs of equation (1.6) is obtained through the applicationof the Weierstrass Gauss transform [8]ey ∂2xf(x) =12√pi y+∞∫−∞e−(x−ξ)24 y f(ξ) dξ (1.7)which allows the derivation of the generating function (1.6) according to the fol-lowing expression∞∑n=0tnn!H2n(x, y) =1√1− 4 y tex2t1−4 y t ,|t| < 14 |y|(1.8)which is sometimes called Doetsch rule [5].The procedure we have adopted to derive equation (1.8) can be generalized toget the following generalization of the Doetsch rule∞∑n=0tnn!H2n+l(x, y) = ey∂2x(et x2xl)=1√1− 4 y tex2t1−4 y tHl(x√1−4 y t , y)(1− 4 y t) l2,|t| < 14 |y| .(1.9)It is evident that the operational method exploited so far can be extended to higherorder lacunary generating functions and we will show, in the forthcoming sections,that it may become a fairly powerful tool once complemented with a notation ofumbral nature.18 G. DATTOLI, B. GERMANO, M.R. MARTINELLI, P.E. RICCI2. Umbra and Hermite polynomialsIn a previous investigation we have shown that symbolic methods seem to betailor suited to deal with the properties of Bessel functions and Laguerre polyno-mials as well [2]. The methods we have developed largely rely on techniques ofumbral type and therefore in order of treating Hermite polynomials we introducethe following “umbra”hˆryϕ0 =yr2 r!Γ(r2 + 1) ∣∣∣cos(rpi2)∣∣∣ . (2.1)Which reduces, for y = 12 , to an analogous quantity defined in [6]. By such anotation we can redefine the Hermite polynomials asHn(x, y) =(x+ hˆy)nϕ0 . (2.2)According to equation (2.2) the Hermite polynomials are reduced to the n-th powerof a binomial. All the relevant properties can be obtained by handling equation(2.2) by means of elementary algebraic tools.The exponentiation of the umbra hˆy will be particularly important in the presentcontext, we note therefore thatehˆy zϕ0 =∞∑r=0(hˆy z) rr!ϕ0 = ey z2 (2.3a)andehˆ2y zϕ0 =1√pi+∞∫−∞e−ξ2+2 ξhˆy√zϕ0dξ =1√1− 4 y z ,|z| < 14 |y| .(2.3b)Which is just a consequence of equation (2.3a), if we note thatehˆ2yzϕ0 =1√pi+∞∫−∞e−ξ2+2 ξ hˆ y√z dξ ϕ0 =1√pi+∞∫−∞e−ξ2+2 ξ hˆy√zϕ0 dξ=1√pi+∞∫−∞e−ξ2(1−4 y z) dξ.(2.4)The Doetsch formula and its extension can be therefore derived by the use of theidentity∞∑n=0tnn!H2n(x, y) = e(x+hˆy)2tϕ0 =1√pi+∞∫−∞e−ξ2+2 (x+hˆy)√tξϕ0 dξ . (2.5)LACUNARY GENERATING FUNCTIONS OF HERMITE POLYNOMIALS 19Let us now consider the same problem from a different point of view and write∞∑n=0tnn!H2n(x, y) = e(x+hˆy)2tϕ0 = ex2tehˆ2y t+2 hˆy x tϕ0= ex2t∞∑r=0hˆryr!Hr(2x t, t)ϕ0 = ex2t∞∑r=0yrr!H2r(2x t, t)(2.6)which has been derived by using equation (1.1b), by comparison with the Doetschrule equation (2.6) yields the operational identityehˆ2y t+2 hˆy x tϕ0 =1√1− 4 y te4 y (t x)21−4 y t . (2.7)Before going further let us note that the third order Hermite polynomials [3]H(3)n (x, y, z) = n![n3 ]∑r=0zrHn−3 r(x, y)r! (n− 3 r)! (2.8)can be defined through the generating function∞∑n=0tnn!H(3)n (x, y, z) = ex t+y t2+z t3 . (2.9)On account of equation (2.2) we can write the triple lacunary Hermite generatingfunction as∞∑n=0tnn!H3n(x, y) = et (x+hˆy)3ϕ0 . (2.10)Which, according to equation (2.9), allows the following conclusion∞∑n=0tnn!H3n(x, y) = et x3∞∑r=0hˆryr!H(3 )r (3x2t, 3x t2, t)ϕ0= et x3∞∑r=0yrr!H(3 )2 r (3x2t, 3x t2, t) .(2.11)Which can be worded as it follows: the triple lacunary generating function of secondorder Hermite polynomials, can be expressed in terms of double lacunary generatingfunction of third order Hermite polynomials.3. The associated Hermite polynomialsIn analogy with the case of Laguerre polynomials [2] we introduce the associatedHermite polynomials, which, according to the present formalism, readHn(x, y|p) = hˆpy(x+ hˆy)nϕ0= n!n∑r=0(r + p)!yr+p2 xn−rΓ(r+p2 + 1)(n− r)!∣∣∣cos((r + p)pi2)∣∣∣ . (3.1)They cannot be identified with the generalized heat polynomials [8], and, withinthe present context, deserve a separate treatment.20 G. DATTOLI, B. GERMANO, M.R. MARTINELLI, P.E. RICCIAccording to the previous definition we can “state” the following index-duplicationformulaH2n(x, y) = (hˆy + x)n(hˆy + x)nϕ0= n!n∑s=0xn−s(n− s)! s!Hn(x, y|s)(3.2)and argument-duplication formulaHn(2x, y) =[(hˆy2+ x)+(hˆy2+ x)]nϕ0=n∑s=0(ns) s∑r=0(sr)xr2s−rHn−s(x,y2|s− r).(3.3)It is furthermore easily checked thatxn =[(x+ hˆy)− hˆy]nϕ0 =n∑r=0(−1)r(nr)Hn−r(x, y|r) (3.4)and thatHn+m(x, y) = (hˆy + x)m (hˆy + x)nϕ0=m∑r=0(mr)xm−rHn(x, y|r) .(3.5)The last identity is a reformulation of the Nilsen theorem, concerning the sum ofthe indices of Hermite polynomials [1]. The previous results (3.2)-(3.5) occur inthe literature in different forms [4] and in the concluding section we will commenton the differences between the ordinary formulation and the ones presented in thispaper.Even though not explicitly mentioned the Hermite umbra can be raised to anyreal power, and this allows noticeable freedom in guessing possible generalizations.A fairly direct example is provided by the following extensionHn(x, y |β;α) = hˆβy(x+ hˆαy)nϕ0 . (3.6)Yielding a family of polynomials with generating function∞∑n=0tnn!Hn(x, y |β;α) = ex tyβ2 eα, β(yα2 t),eα, β(x) =∞∑r=0Γ(α r + β + 1)xrΓ(α r+β2 + 1)r!∣∣∣∣cos(α r + β2 pi)∣∣∣∣ . (3.7)Their properties can easily be studied and they are framed within the context ofthe Sheffer family, they can accordingly be defined through the operational ruleHn(x, y |β;α) = yβ2 eα, β(yα2 ∂x)xn (3.8)In the following part of the paper we will discuss further elements characterizingthe usefulness of the umbral point of view to the theory of Hermite polynomials.LACUNARY GENERATING FUNCTIONS OF HERMITE POLYNOMIALS 214. Umbra, Higher order Hermite polynomialsand final commentsThe Hermite polynomials, defined through the operational identity [3]H(m)n (x, z) = ez ∂mx xn (4.1)are specified by the seriesH(m)n (x, z) = n![ nm ]∑r=0xn−mrzr(n−mr)! r! . (4.2)They can reduced to the n-th power of a binomial by introducing the umbramhˆryϕ0 =yrm r!Γ(rm + 1)Am,rAm,r =(1, r = mp0, otherwhise), p ≡ integer .(4.3)Which allows to define them asH(m)n (x, z) = (mhˆz + x)nϕ0 . (4.4)It is clearly evident that not too much effort is necessary to study the relevantproperties, which can be derived using the same procedure adopted for the secondorder case.We can combine the Hermitian umbra to get further generalizations, as for thethree variable third order Hermite polynomials, which, according to the previousformalism, can be defined asH(3)n (x, y, z) = (3hˆz + 2hˆy + x)nϕ0,zϕ0,y . (4.5)Thereby we findH(3)n (x, y, z) =n∑s=0(ns)(3,2)hˆsz, yxn−sϕ0,zϕ0,y,(3,2)hˆsz, y =s∑r=0(sr)3hˆs−rz 2hˆry .(4.6)The extension of the method to bilateral generating functions is quite straightfor-ward too. We consider indeed the generating functionG(x, y; z, w| t) =∞∑n=0tnn!Hn(x, y)Hn(z, w)=∞∑n=0tnn!(hˆy + x)nHn(z, w)ϕy,0= e(hˆy+x) z t+[(hˆy+x) t]2wϕy,0(4.7)the use of our technique yieldsG(x, y; z, w| t) = 1√1− 4yt2we(x2w+y z2) t2+x t z1−4 y t2w . (4.8)Exotic generating functions involving e.g. products of Laguerre and Hermite poly-nomials can also be obtained and will be discussed elsewhere.22 G. DATTOLI, B. GERMANO, M.R. MARTINELLI, P.E. RICCIThe use of umbral methods looks much promising to develop a new point of viewon the theory of special polynomials and of special functions as well.Just to provide a flavor of the directions along which may develop future specu-lations, we consider the definition of the following umbraHˆnx,yϕ0 = Hn(x, y) . (4.9)Accordingly we can writeet Hˆx,yϕ0 = ex t+y t2 (4.10)and also∞∑n=0tnn!H2n(x, y) = et Hˆ2x,yϕ0=1√pi+∞∫−∞e−ξ2+2 ξ Hˆx,y√tdξ ϕ0=1√pi+∞∫−∞e−(1−4 y t) ξ2+2 ξ x√tdξ =1√1− 4 y tex2t1−4 y t .(4.11)Let us now consider the following integralI (x, y) =+∞∫−∞e−z2Hˆx,− y dz ϕ0 (4.12a)equivalent toI(x, y) =+∞∫−∞e−y z4−x z2d z =+∞∫0e−s x−s2ys−12 ds . (4.12b)The same result will be now derived using a symbolic procedure, based on theapplication of the Ramanujan master theorem [7], which allows to write the formalsolution of the integral in equation (4.12a) according to the identityI (x, y) =√pi Hˆ− 12x,−yϕ0 . (4.13)We are therefore faced with the necessity of specifying the meaning of the Hermiteumbra raised to a negative fractional power.The use of standard Laplace transform methods yieldsHˆ− 12x,−yϕ0 =1Γ(12) +∞∫0e−sHˆx,− ys−12 dsϕ0 =1√pi+∞∫0e−s x−s2ys−12 ds (4.14)which correctly reproduces equation (4.12b). This is quite a significant result, whichensures that the formalism has a very high level of flexibility.It is also interesting to note that the “Fourier” transformFˆ (f ; k, β)ϕ0 =1√2pi+∞∫−∞f(x) ei xHˆk, βdxϕ0 (4.15)can be viewed as a kind of Gabor transform.LACUNARY GENERATING FUNCTIONS OF HERMITE POLYNOMIALS 23The technique we have discussed in the paper is tightly bound to the method ofquasi-monomials developed in [9], provided that we make the following identification(x+ hˆy)→ (x+ 2 y ∂x) . (4.16)The differential operator on the left is used to define the Hermite polynomials asHn(x, y) = (x+ 2 y ∂x)n1 . (4.17)There are certain advantages offered by the umbra method with respect to themonomiality technique, which are all associated with the fact that in the formercase one deals with commuting operators.References[1] P. Appell, J. Kampe´ de Fe´riet, “Fonctions Hyperge´ome´triques et Hypersphe´riques :Polynoˆmes d’Hermite”, Gauthier-Villars, Paris, 1926.[2] D. Babusci, G. Dattoli, K. Go´rska, Symbolic calculus and integrals of Laguerre polynomials,arXiv:1202.1976v1 [math.CA] 9 Feb. 2012.[3] G. Dattoli, P. L. Ottaviani, A. Torre, L. Va´zquez, Evolution operator equations: Integrationwith algebraic and finite-difference methods.Applications to physical problems in classical andquantum mechanics and quantum field theory, Riv. Nuovo Cimento Soc. Ital. Fis: (Ser. 4)20 (1997), 1–113.[4] G. Dattoli, A. Torre, M. Carpanese, Operational rules and arbitrary order Hermite generatingfunctions. J. Math. Anal. Appl. 227 (1998), n. 1, 98–111.[5] G. Doetsch, Integraleigenschaften der Hermiteschen Polynome, Math. Z. 32 (1930), 557–559.[6] I. M. Gessel , P. Jaywant, A triple lacunary generating function for Hermite polynomials,arXiv: math/0403086v2 [math.CO] 22 Jul 2005.[7] G. H. Hardy, “Ramanujan. Twelve Lectures on Subjects Suggested by his Life an Work”,Cambridge University Press, Cambridge, 1940.[8] I.I. Hirschman, D. V. Widder, “The Convolution Transform”. Princenton University press,Princeton, N. J., 1955. x + 268 pp.[9] P. E. Ricci, I. Tavkhelidze, An introduction to operational techniques and special polynomials.Differential equations and their applications. J. Math. Sci. (N. Y.) 157 (2009), n.1, 161–189.[10] D. V. Widder, “The Heat Equation”, Pure and Applied Mathematics, Vol.67, Academic Press[Harcourt Brace Jovanovich, Publishers], New York-London, 1975. xiv + 267 pp.Giuseppe DattoliUnita` Tecnico Scientifica Tecnologie Avanzate, ENEA-Centro Ricerche Frascati, C.P.65, Via Enrico Fermi 45, I-00044, Frascati, Roma, ItaliaE-mail address: dattoli@frascati.enea.itBruna GermanoSapienza - Universita` di Roma, Dipartimento di Scienze di Base e Applicate per l’Ingegneria,Via A. Scarpa, 14 - 00161 Roma, ItaliaE-mail address: bruna.germano@sbai.uniroma1.itMaria Renata MartinelliSapienza - Universita` di Roma, Dipartimento di Scienze di Base e Applicate per l’Ingegneria,Via A. Scarpa, 14 - 00161 Roma, ItaliaE-mail address: martinelli@dmmm.uniroma1.itPaolo Emilio RicciInternational Telematic University UniNettuno, Corso Vittorio Emanuele II, 39 - 00186Roma, ItaliaE-mail address: paoloemilioricci@gmail.com

Lacunary generating functions of Hermite polynomials and symbolic methods

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Lacunary generating functions of Hermite polynomials and symbolic methods

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