We develop a new method of umbral nature to treat blocks of Her

mite and of Hermite like poly-

nomials as independent algebraic quantities.  The Calculus

we propose allows the formulation of

a number of ”practical rules” allowing significant simplific

ations in computational problem

Dattoli, G

Germano, B

Martinelli, M. R.

English

We develop a new method of umbral nature to treat blocks of Her
mite and of Hermite like poly-
nomials as independent algebraic quantities.  The Calculus
we propose allows the formulation of
a number of ”practical rules” allowing significant simplific
ations in computational problem

Dattoli, Giuseppe

Germano, Bruna

Licciardi, Silvia

Martinelli, Maria Renata

Archivio della ricerca- Università di Roma La Sapienza

Hermite CalculusGiuseppe Dattoli, Bruna Germano, Silvia Licciardiand Maria Renata MartinelliAbstract We develop a newmethod of umbral nature to treat blocks of Hermite andof Hermite like polynomials as independent algebraic quantities. The Calculus wepropose allows the formulation of a number of “practical rules” yielding significantsimplifications in computational problems involving integrals and partial differentialequations as well. The procedure we adopt is particularly useful to enter more deeplyin the algebraic structure of Hermite polynomials. It provides indeed a tool allowinga generalization of the recently introduced geometrical point of view to the interplaybetween ordinary monomials and Hermite polynomials.1 IntroductionIn this paper we deal with a protocol, which will be referred as Hermite calculus,useful to treat computations involvingHermite polynomials and their generalizationsas well.G. Dattoli (B) · S. LicciardiENEA - Frascati Research Center, Via Enrico Fermi 45, 00044 Frascati, Rome, Italye-mail: giuseppe.dattoli@enea.itB. Germano · M.R. MartinelliDepartment of Methods and Mathematic Models for Applied Sciences,University of Rome, La Sapienza, Via A. Scarpa, 14, 00161 Rome, Italye-mail: bruna.germano@sbai.uniroma1.itM.R. Martinellie-mail: martinelli@dmmm.uniroma1.itS. LicciardiDepartment of Mathematics, University of Catania, Via Santa Sofia,64, 95125 Catania, Italye-mail: silvia.licciardi@dmi.unict.it© Atlantis Press and the author(s) 2017J. Gielis et al. (eds.), Modeling in Mathematics,Atlantis Transactions in Geometry 2, DOI 10.2991/978-94-6239-261-8_44344 G. Dattoli et al.To give a flavour of the techniques we will employ we consider the integralI (α,β, γ) =∫ ∞−∞e−(α+β)x2−γxdx (1)which can be evaluated with ordinary means, thus gettingI (α,β, γ) =√πα + β eγ24(α+β) (2)We will test the formalism, we are going to decribe, by using such a benchmarkand restyle Eq. (1) as it followsI (α,β, γ) =∫ ∞−∞e−αx2−hˆ(γ,−β)xdx (3)where we have introduced the notatione−hˆ(γ,−β)x =∞∑r=0(−x)rr ! hˆr(γ,−β) =∞∑r=0(−x)rr ! Hr (γ,−β) (4)based on the use of the umbral identity [1, 2]1hˆr(γ,−β) = Hr (γ,−β),Hr (x, y) = r ![ r2 ]∑s=0xr−2s ys(r − 2s)!s!(5)In the integral in Eq. (4) we have treated the term which can be expended in termsof Hermite polynomials as a single block and we have enucleated the variable xraised to the first power.We will now follow a prescription according to which the operator hˆ is treated asan ordinary algebraic quantity.According to the ordinary rules for the Gaussian integrals we can write [2, 3]I (α,β, γ) =√παehˆ2(γ,−β)4α =√πα∞∑r=01r !(hˆ2(γ,−β)4α)r(6)which provides us with the correct result for the problem we are studying. Theapplication of the previous prescription yields, indeed, if∣∣∣∣βα∣∣∣∣ < 11We have used the umbral notation in a rather inaccurate way, without specifying on which spacethe operators are acting. For further comments and an appropriate discussion of the formal content,see the second of Ref. [2].Hermite Calculus 45√πα∞∑r=01r !(hˆ2(γ,−β)4α)r==√πα∞∑r=01r !H2r(γ2√α,− β4α)=√πα + β eγ24(α+β)(7)which is obtained after using the identity [2]∞∑r=0trr !H2r (x, y) =1√1 − 4yt ex2 t1−4yt (8)The result in Eq. (7), evidently correct, yields some confidence on the reliability ofthe formalism, which is based on a rather wild use of the umbral formalism.2 The Umbral Method and Generalized HermitePolynomialsEven though not explicitly stated, the umbral operator defined in Eq. (5) satisfies theidentity2hˆm hˆr = hˆm+r (9)where the powers m and r are not necessarily real integers. It is also fairly natural toset∂hˆ hˆr = r hˆr−1 = r Hr−1(γ,−β) (10)Since the following recurrence holds∂γHr (γ,−β) = r Hr−1(γ,−β) (11)the “derivative” operator can therefore be identified with∂hˆ → ∂γ (12)Furthermore sincehˆhˆr = hˆr+1 = Hr+1(γ,−β) (13)and, on account of the recurrence,Hr+1(γ,−β) = γHr (γ,−β) − 2 β r Hr−1(γ,−β) (14)2The subscript (γ,−β) has been omitted because the identity holds for hˆ operators with the samebasis, hereafter it will be included whenever necessary.46 G. Dattoli et al.we can also conclude that hˆ itself can be identified with the differential operatorhˆ = γ − 2 β ∂γ (15)It is also worth noting that∂rx e−hˆx = (−1)r hˆr e−hˆx = (−1)r∞∑n=0(−x)nn! hˆn+r == (−1)r∞∑n=0(−x)nn! Hn+r (γ,−β)(16)and, according to the identity∞∑n=0tnn!Hn+l(x, y) = Hl(x + 2yt, y)ext+yt2 (17)we can establish the rulehˆr e−hˆx = Hr (γ + 2 β x,−β)e−(γx+βx2) (18)We can now make a step further by defining the integralI (γ,β) =∫ ∞−∞e−hˆx2dxe−hˆx2 = e−(γx2+βx4)(19)which, after applying the prescription that hˆ can be treated as an ordinary algebraicquantity, writesI (γ,β) = √πhˆ− 12 (20)which makes sense only if we can provide a meaning for hˆ− 12 , the most naturalconclusion is that they can be understood as factional order Hermite, which for ourpurposes can be defined as it follows [4]Hν(x,−y) = y ν2 ex24y√π∫ ∞0e−t24 tν cos(x2√yt − π2ν)dt (21)or asHν(x,−y) = (ν + 1)∞∑r=0xν−2r (−y)r(ν + 1 − 2r)r ! , x >> y (22)which has however a limited range of convergence.Hermite Calculus 47The correctness of Eq. (20) can be readily proved by a numerical check, involvingeither the definitions (21) and (22).We will comment, later in this paper, on the extension of the Hermite polynomialsto non-integer index.Let us now consider the following repeated derivatives∂ nx e−hˆx2 = (−1)nHn(2 hˆ x,−hˆ)e−hˆx2 == (−1)nn![ n2 ]∑r=0(−1)r (2x)n−2r(n − 2r)!r !(hˆn−r e−hˆx2) (23)Thus, getting, on account of Eq. (18),∂ nx e−hˆx2 = (−1)nn![ n2 ]∑r=0(−1)r (2x)n−2r(n − 2r)!r ! Hn−r (γ + 2 βx2,−β)e−(γx2+βx4) (24)in accordance with∂ nx e−(γx2+βx4) = H (4)n (−2 γx − 4 βx3,−γ − 6 βx2,−4 βx,−β)e−(γx2+βx4)(25)In Ref. [5] the following integralJ (a, b, c) =∫ ∞−∞e−(ax4+bx2+cx)dx,Re(a) > 0(26)has been considered, within the framework of problems regarding the non-perturbative treatment of the anharmonic oscillator. A possible perturbative treat-ment is that of settingJ (a, b, c) =∞∑n=0(−1)nn! gn(a) [Hn(c,−b) + Hn(−c,−b)] ,gn(a) =∫ ∞0xne−ax4dx = 14a−n+14 (n + 14) (27)which, as noted in [5], is an expansion with zero radius of convergence in spite ofthe fact that J (a, b, c) is an entire function for any real or complex value of b, c.The use of our point of view allows to writeJ (a, b, c) =∫ ∞−∞e−hˆ(b,−a)x2−cxdx =√πhˆec24hˆ = √π∞∑s=01s!( c2)2shˆ−(s+12 ) (28)48 G. Dattoli et al.We have omitted the subscript (b,−a) in the r.h.s. of Eq. (28) to avoid a cumbersomenotation. The meaning of the operator hˆ raised to a negative exponent is easilyunderstood ashˆ−(s+12 ) = H−(s+ 12 )(b,−a) (29)where the negative indexHermite polynomials are expressed in terms of the paraboliccylinder functions Dn according to the identity [6]H−n(x,−y) = (2y)− n2 e x28y D−n(x√2y)(30)The use of Eq. (30) in Eq. (28) finally yields the same series expansion obtainedin Ref. [5]J (a, b, c) = √π∞∑s=01s!( c2)2s(2a)−12 (s+ 12 ) eb28a D−(s+ 12 )(b√2a)(31)which is convergent for any value of b, c and a > 0.3 Final CommentsRegarding the use of non-integer Hermite polynomials it is evident that the definitionadopted in Eq. (21) can be replaced by the use of the parabolic cylinder function, itis therefore worth noting that the use of the properties of the D functions allows thefollowing alternative form for Eq. (20) (see Ref. [7])I (γ,β) = √π (2 β)− 14 e γ28β D− 12(γ√2β)==√γ2√2β(2 β)−14 eγ28β K 14(γ28β);D− 12 (z) =√z2πK 14(14z2)(32)where Kν(z) is a modified Bessel function of the second kind.A further example of application of the method developed so far is provided by∫ ∞0e−(βx2n+γxn)dx =∫ ∞0e−hˆ(γ,−β)xndx = 1n(1n)hˆ− 1n(γ,−β) == 1n(1n)(2β)−12n eγ28β D− 1n(γ√2β) (33)Hermite Calculus 49In a forthcoming more detailed note we will extend the method to cases involvinghigher order Hermite polynomials. Just to provide an idea of the extension of thetechnique,we note that the use of this family of polynomials allows to cast the integralin Eq. (28) in the formJ (a, b, c) =∫ ∞−∞e 4 hˆ(−c,−a)x−bx2dx =√πbe( 4 hˆ(−c,−a))24b (34)where (4hˆ(−c,−a))n = H (4)n (−c,−a) = (−1)nn![ n4 ]∑r=0cn−4r (−a)r(n − 4r)!r ! (35)with H (4)n (c,−a) being a fourth order Hermite Kampé de Fériét [8] polynomial.The series expansion of the right hand side of Eq. (34) in terms of fourth orderHermite converges in a much more limited range than the series (31) and has beenproposed to emphasize the possibilities of the method we have proposed so far.According to our formalism the Pearcey integral, widely studied in optics, withinthe framework of diffraction problems [9], is easily reduced to a particular case ofEq. (26), namelyJ (1, x,−iy) =∫ ∞−∞e−(t4+xt2)+iyt dt =√πhˆ(x,−1)e− y24hˆ(x,−1) (36)and can be expressed in terms of parabolic cylinder functions, as indicated before. Itis perhaps worth stressing that, in the literature a converging series for the Pearceyintegral is given in the form [10]J (1, x,−iy) =∫ ∞−∞e−t4−hˆ(iy,−x)t dt ==∫ ∞0e−t4(ehˆ(iy,−x)t + ehˆ(−iy,−x)t)dt = 2∞∑n=0(−1)ng2n(1)a2n(x, y) (37)witha0(x, y) = 1,a1(x, y) = y,an(x, y) = 1n(y an−1(x, y) + 2 x an−2(x, y))(38)which is reconciled with our previous result, in terms of two variable Hermite poly-nomials, provided that one recognizes50 G. Dattoli et al.J (1, x,−iy) =∞∑n=0(−1)nn! gn(1) [Hn(−iy,−x) + Hn(iy,−x)] (39)In this letterwe have provided somehint on the use of theHermite calculus to studyintegral formswith specific application in different field of research. In a forthcomingmore detailed note we will show how the method can be extended to a systematicinvestigation of the Voigt functions and to the relevant generalizations [11].A further important application is the use of the method within the framework ofevolutive PDE. To start, we consider the following straightforward example:∂t F(x, t) =(α∂x + β∂2x)F(x, t)F(x, 0) = f (x) (40)whose (formal) solution is easily obtained asF(x, t) = e(αt)∂x+(βt)∂2x f (x) (41)The use of the formalism developed so far allows to write the rhs of Eq. (41) in theformF(x, t) = ehˆ(αt, βt)∂x f (x) (42)by the use of standard exponential rules we obtainF(x, t) = f(x + hˆ(αt, βt))(43)which is still a formal solution unless we provide a meaning for the rhs of Eq. (43).Let us therefore use the Fourier transform method to writef(x + hˆ(αt, βt))= 1√2π∫ ∞−∞f˜ (k)eikx+hˆ(αt, βt)ikdk == 1√2π∫ ∞−∞f˜ (k)eik(x+αt)−k2βt dk(44)which is a kind of Gabor transform [12]. It is evident that the same result can beobtained with ordinary means, we have used this example to prove the correctnessand flexibility of the method we propose.Let us now specialize the result to the case f (x) = xn and writef(x + hˆ(αt, βt))=(x + hˆ(αt, βt))n =n∑s=0(ns)xn−s hˆs(αt, βt) == Hn(x + αt, βt)(45)Hermite Calculus 51which is just the derivation from a different point of view of the following operationalidentity [13]eκ∂x+λ∂2x xn = Hn(x + κ,λ) (46)We want also to emphasize that the methods we have just outlined offer furtherpossibility of speculation on the nature of the hˆ operator itself, which can be furtherexploited to introduce nested Hermite polynomials. Its umbral nature and the asso-ciated flexibility allows indeed noticeable degree of freedom therefore the followingidentity derives naturally from the previous prescriptionehˆ(α, β)∂2x xn = Hn(x, hˆ(α, β))== n![ n2 ]∑r=0xn−2r Hr (α, β)(n − 2r)!r !(47)This is what we define nested Hermite, for obvious reasons. The geometrical natureof the operator itself can be understood as discussed in [14] and will be furthercommented elsewhere.Before closing the paper wewant to underline that the possibilities for the applica-bility of the integration method discussed in this letter arise if, inside the integrand,an exponential generating function is recognized.To clarify this point we note that the integralf (a, b, c) =∫ ∞−∞e−ax2+√x2+bx+cdx,b2 − 4c < 0a > 1(48)can be written asf (a, b, c) =√πaeRˆ24a =√πa∞∑n=01n!(14a)nR2n(b, c) (49)provided thate√x2+bx+c = eRˆx =∞∑n=0xnn! Rˆn,Rˆn = Rn(b, c)(50)where Rn(b, c) are polynomials of the parameter b, c.Even though such a polynomials expansion can be obtained using different proce-dure, we have tested the validity of our ansatz using the following integral definition52 G. Dattoli et al.Rm(b, c) = m!2π∫ 2π0e−imφe√e2iφ+beiφ+cdφ (51)which has been used to benchmark the identity (49), with the full numerical integra-tion of (48). Further comments will be provided elsewhere.References1. E.W. Weisstein, Hermite Number, from MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/HermiteNumber.html2. G. Dattoli, B. Germano, M.R. Martinelli, P.E. Ricci, Lacunary generating functions of Hermitepolynomials and symbolic methods. Ilirias J. Math. vol. 4(1), 16–23 (2015). ISSN: 2334-6574,http://www.ilirias.com; (G. Dattoli, E. Di Palma, E. Sabia, K. Gorska, A. Horzela, K. PensonOperational Versus Umbral Methods and the Borel Transform (2015). arXiv:1510.01204v1[math.CA])3. H.M. Srivastava, L.Manocha,A treatiseOnGenerating Functions. Bull. Am.Math. Soc. (N.S.),vol. 19(1), 346–348 (1988)4. R. Hermann, Fractional Calculus: An Introduction for Physicists, 2nd edn. (World Scientific,Singapore, 2014)5. J. Bohacik, P. Augustin, P. Presnajder, Non-perturbative anharmonic correction to Mehler’spresentation of the harmonic oscillator propagator. Ukr. J. Phys. 59, 179 (2014)6. M. Abramovitz, I.A. Stegun, (eds), Parabolic Cylinder Function, Chap. 19 in Handbook ofMathematical Functions with Formulas, Graphs and Mathematical Tables, 9th Printing (NewYork, Dover, 1972), pp. 685–7007. E.W. Weisstein, Parabolic Cylinder Function, from MathWorld–A Wolfram Web Resource,http://mathworld.wolfram.com/ParabolicCylinderFunction.html8. P. Appél, J. Kampé de Fériét, Fonctions hypérgeométriques and Hypérspheriques; polinómesdHermite (Gauthier-Villars, Paris, 1926)9. J.L. López, P.J. Pagola, arXiv:1601.03615 [mat.NA]10. M.V. Berry, C.J. Howls, Integrals with coalescing saddles, in NIST Handbook of MathematicalFunctions (Cambridge University Press, Cambridge, 2010), pp. 775793. (Chapter 36)11. M.A. Pathan, Scientia, Series A: Mathematical Sciences, vol. 12(9) (2006)12. Gabor Analysis andAlgorithms Theory andApplications, ed. ByH.G. Feichtinger, T. Strohmer,Birkhuser (1998)13. D. Babusci, G. Dattoli, M. Del Franco, Lectures on Mathematical Methods for Physics,RT/2010/58/ENEA (2010)14. M. Artioli, G. Dattoli, The Geometry of Hermite Polynomials. Wolfram Demon-strations Project, Published: March 4 (2015), http://demonstrations.wolfram.com/TheGeometryOfHermitePolynomials/

Hermite Calculus

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Hermite Calculus

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