On integrals involving Hermite polynomials

On integrals involving Hermite polynomials

Applied Mathematics Letters 25 (2012) 1157–1160 Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: www...

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Applied Mathematics Letters 25 (2012) 1157–1160

Contents lists available at SciVerse ScienceDirect

Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

On integrals involving Hermite polynomials D. Babusci a,∗ , G. Dattoli b , M. Quattromini b a

INFN - Laboratori Nazionali di Frascati, via E. Fermi 40, I-00044 Frascati, Italy

b

ENEA - Centro Ricerche Frascati, via E. Fermi 45, I-00044 Frascati, Italy

article

abstract

info

Article history: Received 22 June 2011 Received in revised form 17 February 2012 Accepted 17 February 2012

We show how the combined use of the generating function method and of the theory of multivariable Hermite polynomials is naturally suited to evaluate integrals of Gaussian functions and of multiple products of Hermite polynomials. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Integrals Hermite polynomials Generating function

Integrals of Hermite polynomials and Gaussians are ubiquitous in problems concerning classical [1] and quantum optics [2] and in quantum mechanics as well [3]. They are exploited to calculate the optical mode overlapping and transition rates between quantum eigenstates of the harmonic oscillator. Albeit they are well known and reported in a variety of standard mathematical tables (see, for example, Ref. [4]), a general method allowing the direct evaluation of these integrals has not been developed. In this letter we fill the gap describing a unifying method, amenable for further generalizations, that provides a kind of automatic procedure for the evaluation of this class of integrals, even in the case that they assume a particularly complicated form. The generating function method is often exploited to derive the analytic form of integrals of the type [5] ∞

 In =

2 dx Hn (a x + b, y)e −c x +α x

(1)

−∞

where Hn (x, y) = n!

[ n/2]

xn−2 k yk

k=0

(n − 2 k)! k!

(2)

is the two variable Hermite polynomial with generating function [6] ∞ n  t

n! n =0

2

H n ( x , y ) = ex t + y t .

(3)

By taking into account this identity, from Eq. (1) we obtain ∞ n  t

n! n =0



In = e

b t +y t 2





2 dx e −c x +(a t +α) x ,

−∞

Corresponding author. E-mail address: [email protected] (D. Babusci).

0893-9659/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.02.043

(4)

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D. Babusci et al. / Applied Mathematics Letters 25 (2012) 1157–1160

and, thus, we have reduced our problem to the evaluation of a trivial Gaussian integral, which yields ∞ n  t n =0

n!

√     2  π α a a2 α In = √ exp + b+ t + y+ t2 . 4c

c

2c

(5)

4c

The use of the generating function (3) allows us to write the integral (1) as follows



π



In = √ exp c

α2



 Hn

4c

b+

αa 2c

,y +

a2



4c

.

(6)

Moreover, since Hn (x, y) = exp y ∂x2 xn ,





(7)

we can write In in the following operational form

√  2    π α a2 αa In = √ exp exp y+ ∂b2 + ∂ b bn . 4c

c

4c

(8)

2c

Let us now consider the integral ∞

 m In

=

2 dx xm Hn (a x + b, y)e −c x +α x ,

(9)

−∞

that occurs in some problems involving quantum harmonic oscillator, or, in classical optics, in the evaluation of overlapping of Gauss–Hermite beams. It can be cast in the form m In

= ∂αm In ,

(10)

from which, taking into account the identities

∂xk Hn (x, y) =

n!

(n − k)!

∂xk e β x = Hk (2 β x, β) e β x , 2

Hn−k (x, y)

2

(11)

one finds

√ π



exp m In = √ c

α2



4c

α

 Hm,n

,

1

2c 4c

;b +

αa 2c

,y +

a2 4c

|

a 2c



,

(12)

where we have introduced the two-index Hermite polynomials [6,7] defined as follows Hm,n (x, y; w, z | τ ) =

min(m,n)



m ! n!

k=0

(m − k)! (n − k)! k!

τ k Hm−k (x, y) Hn−k (w, z ).

(13)

The relevant generating function writes ∞    um v n Hm,n (x, y; w, z | τ ) = exp x u + y u2 + v w + z v 2 + τ u v , m ! n! m,n=0

(14)

and, thus, using the same procedure as before, we find for integrals of the type ∞

 Im,n =

2 dx Hm (a x + b, y) Hn (c x + d, z ) e −f x +α x

(15)

−∞

the explicit form



π

Im,n = √

f

 exp

α2 4f



Hm,n (¯x, y¯ ; w, ¯ z¯ | τ ) ,

(16)

where we put x¯ = b +

a 2f

α,

y¯ = y +

a2 4f

,

w ¯ =d+

c 2f

α,

z¯ = z +

c2 4f

,

τ=

ac 2f

.

By taking into account that 2 Hm,n (x, y; w, z | τ ) = exp y ∂x2 + w ∂z2 + τ ∂xz xm y n





(17)

D. Babusci et al. / Applied Mathematics Letters 25 (2012) 1157–1160

1159

we obtain the generalization of Eq. (8) as

√ π



Im,n = √ exp f

α2



¯ ∂z¯2 + τ ∂x¯2z¯ x¯ m y¯ n . exp y¯ ∂x¯2 + w 



4f

(18)

The operatorial method can be applied from the very beginning and the integral (16) can be written as Im,n = exp y ∂b2 + z ∂d2 Im,n ,





(19)

where





Im,n =

2 dx (a x + b)m (c x + d)n e −f x +α x

−∞

√   2  π c2 a2 α = √ exp Hm,n x¯ , ; w, ¯ |τ . 4f

f

4f

(20)

4f

This approach may simplify the computation of integrals involving products of more than two Hermite polynomials. It can now be shown that for the integral ∞

 p Im,n =

2 dx xp Hm (a x + b, y) Hn (c x + d, z ) e −f x +g x ,

(21)

−∞

an identity analogous to (10) holds p Im,n

= ∂αp Im,n .

(22)

The use of the identities m! Hm−k,n (x, y; w, z | τ ) (m − k)! n! ∂wk Hm,n (x, y; w, z | τ ) = Hm,n−k (x, y; w, z | τ ), (n − k)!

∂xk Hm,n (x, y; w, z | τ ) =

(23)

allows us to obtain for the integral (21) the following expression p Im,n

√    2  p   π p α 1 α = √ exp Hp−k , R(mk,)n 4f

f

k

k=0

2f

4f

(24)

where R(mk,)n = ∂αk Hm,n (¯x, y¯ ; w, ¯ z¯ | τ )

 =

c

k  k    l a k

2f

l

l=0

m! n!

(m − l)! (n − k + l)!

c

Hm−l,n−k+l (¯x, y¯ ; w, ¯ z¯ | τ ) .

(25)

As a final example, we consider the integral ∞

 In =

dx −∞

(a x + b)n (1 + c x2 )ν

(n < 2 ν, ν ∈ R).

(26)

The combined use of the generating function and Laplace transform methods yields1 ∞ n  t

n! n =0

In =

ebt





Γ (ν)

ds e −s sν−1





2 dx e −s c x +a x t

(27)

−∞

0

which can be finally explicitly worked out as



π

In = √ Hn(ν) c Γ (ν)



b,

a2 4c



,

(28)

where Hn(ν) (x, y) = n!

[ n/2] k=0

Γ (ν − k − 1/2) n−2 k k x y . k! (n − 2 k)!

(29)

1 In performing the sum we do not take into account the condition n < 2 ν , but the consequent result in Eq. (28) must be considered valid only under this hypothesis.

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These polynomials do not belong to the Appell family and the relevant generating function can be written adopting an umbral notation as follows ∞ n  t

n! n =0

Hn(ν) (x, y) = e x t +ˆy t

2

yˆ k = Γ (ν − k − 1/2) yk .

(30)

With this assumption we also get Hn(ν) (x, y) = exp yˆ ∂x2 xn ,





and then most of the properties of this family of polynomials can be derived straightforwardly. References [1] [2] [3] [4] [5] [6] [7]

L.C. Andrews, Special Functions for Engineers and Applied Mathematicians, Mac Millan, New York, 1985. W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1990. N.N. Lebedev, Special Functions and their Applications, Dover, New York, 1972. I.S. Gradshteyn, I.M. Ryzhik, in: A. Jeffrey (Ed.), Table of Integrals, Series, and Products, fifth ed., Academic Press, New York, 1994, p. 708. D. Babusci, G. Dattoli, M. Del Franco, Lectures on mathematical methods for physics, Internal Report ENEA RT/2010/5837. P.E. Appell, J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques: Polynômes d’Hermite, Gauthier-Villars, Paris, 1926. G. Dattoli, J. Comput. Appl. Math. 118 (1–2) (2000) 111–123.

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