fulltext

(1)

IL NUOVO CIMENTO VOL. 112 B, N. 11 Novembre 1997 NOTE BREVI

Two-index two-variable Hermite-Bessel functions

for synchrotron radiation in two-frequency undulators

G. DATTOLIand A. TORRE

ENEA, Dipartimento Innovazione, Divisione Fisica Applicata, Centro Ricerche Frascati C.P. 65, 00044 Frascati, Roma, Italy

(ricevuto il 15 Luglio 1997; approvato il 16 Settembre 1996)

Summary. — The Hermite-Bessel functions with two indices and two variables are

constructed by means of generalized Hermite polynomials. The importance of these functions for the analysis of the radiation emitted by relativistic electrons in exhotic undulators is analyzed.

PACS 02.30.Gp – Special functions.

1. – Introduction

In a previous paper [1] we have introduced the notion of Hermite-Bessel functions (HBF) as functions which exploit the quasi-monomiality property of Hermite poly-nomials. We started indeed from the identities

. / ´ p ×Hen(x) 4nHen 21(x) , m ×Hen(x) 4nHen 11(x) , (1) where p × 4 d dx , m× 4x d dx ( 2 )

and Hen(x) are the Hermite polynomials provided by the generating function [2] ext 2t2O2 4

!

n 40 Q _tn n!Hen(x) (3) to introduce the HBFhJn(x) as hJn(x) 4

!

s 40 Q ₍₂₁₎s_He n 12s(x) s! (n 1s)! 2n 12s (4)

(2)

G.DATTOLIandA.TORRE

1558

which satisfies the recurrences

.

/

´

p× hJn(x) 4 1 2[hJn 21(x) 2hJn 11(x) ] , 2 nhJn(x) 4m×[hJn 21(x) 1hJn 11(x) ] (5a)

and the differential equation

[m×p×m×p× 1m×2

2 n2]hJn(x) 40 . (5b)

In this note we will show that this class of functions can be generalized to the two-variable two-index case, by exploiting the generalized Hermite polynomials provided by the generating function [3, 4]

exTM×t21O2tT_M_×t 4_{n , m}

!

t 1n_t 2m n! m! Hen , m(x1, x2) , (6) where

.

`

/

`

´

x 4

u

x1 x2

v

, _{t 4}

u

t1 t2

v

, M×₄

u

a b b c

v

, a , c D0 , D 4ac2b 2 D 0 (7)

and “T” denotes transpose.

The above polynomials are easily shown to satisfy the recurrences

.

`

/

`

´

P × x1Hen , m(x1, x2) 4nHen 21, m(x1, x2) , P × x2Hen , m(x1, x2) 4mHen , m 21(x1, x2) , M×x1Hen , m(x1, x2) 4Hen 11, m(x1, x2) , M×x2Hen , m(x1, x2) 4Hen , m 11(x1, x2) , (8a) with P 4M×21

u

¯x1 ¯x2

v

, M×4 M×x2

u

¯x1 ¯x2

v

. (8b)

We introduce the HBF specified by the series expansion

hJn , m(x1, x2) 4

!

s 40 Q

!

l 40 Q ₍₂₁₎l 1s l!(m 1l)! s!(n1s)! He_{n 12s, m12l}(x1, x2) 2n 1m12(s1l) , (9)

(3)

TWO-INDEX TWO-VARIABLE HERMITE-BESSEL FUNCTIONS ETC. 1559

which is easily shown to satisfy the recurrences

.

`

/

`

´

2 P×hJn , m(x1, x2) 4

u

hJn 21, m(x1, x2) 2hJn 11, m(x1, x2) hJn , m 21(x1, x2) 2hJn , m 11(x1, x2)

v

, 2 nhJn , m(x1, x2) 4 M×

u

hJn 21, m(x1, x2) 1hJn 11, m(x1, x2) hJn , m 21(x1, x2) 1hJn , m 11(x1, x2)

v

,

u

n 4

u

n m

v

(10a)

and the differential equation (a 41, 2) [ M×xa P × xa M × xa P × xa1 M ×2 xa2 (na) 2 ]nJn , m(x1, x2) 40 . (10b)

Furthermore, it is worth stressing that the function (9) can be derived from the same generating function (6) with the prescription

t K 1 2

u

t12 1 Ot1 t22 1 Ot2

v

. (11a)

This fact is particularly useful to identify hJm , n(x , y) as a member of the class of two-index BF introduced in ref. [5] and playing a crucial role in electromagnetism [4]. We obtain indeed that the argument of the generating function can be written as

1 2(ax11 bx2)(t12 1 Ot1) 1 1 2(bx11 cx2)(t22 1 Ot2) 2 (11b) 21 8[a(t12 1 Ot1) 2 1 c(t22 1 Ot2)21 2 b(t12 1 Ot2)21 2 b(t12 1 Ot1)(t22 1 Ot2) ] . We find therefore that hJm , n(x1, x2) can be written as the following double series:

hJm , n(x1, x2) 4

!

r , q(2) r

hJm 2r2q(r , 2aO2)hJn 1q2r(h , 2cO2) Ir(bO2) Iq(bO2) , (12) where

u

r h

v

4 M ×

u

x1 x2

v

(13)

and hJm , n(x , y) are the HBF defined in ref. [1] and specified by the generating function

!

n 4Q 2Q tn hJn(x , y) 4exp

k

x 2(t 21Ot)1 y 4 (t 21Ot) 2

l

_. (14)

(4)

G.DATTOLIandA.TORRE

1560

Finally, Im(x) is the modified form of the first-kind ordinary BF [2].

The importance of hJm , n(x1, x2) in problems relevant to the emission by charges, whose motion cannot be treated by using the standard dipole approximation and specified by two characteristic frequencies, is understood from the generating function. By setting t14 eiuand t24 eiWin eq. (11b) we obtain

ei(r sin u 1h sin W)1 (1O2)(a sin2u 1c sin2W 12b sin u sin W)₄

!

m , ne

i(mu 1nW)

hJm , n(x1, x2) . (15)

If we set u 4v1t , W 4v2t , where t is the time and v1 , 2two frequencies not prime to each other, we can interpret hJm , n(x1, x2) as the amplitude function for the harmonics generated at mv1and nv2.

Before closing this note, let us emphasize that

1) the modified forms of hJm , n(x1, x2) are obtained from the generating function

e(r cos u 1h cos W)2 (1O2)(a cos2u 1c cos2W 12b cos u cos W)₄

!

m , ne

i(mu 1nW)

hIm , n(x1, x2) ; (16)

2) the two-variable two-index Hermite polynomials are characterized by an adjoint form, thus allowing to define further HBF adjoint to hJm , n(x1, x2) and to

hIm , n(x1, x2).

This aspect of the problem along with a more complete treatment of the theory and of the applications, will be discussed elsewhere.

R E F E R E N C E S

[1] DATTOLIG., TORREA. and MAZZACURATIG., Nuovo Cimento B, 112 (1997) 133.

[2] ANDREWSL. C., Special Functions for Engineers and Applied Mathematicians (Mc Millan, New York) 1985.

[3] HERMITE CH., C.R.AA.S.S., 58 (1864) 93; APPELL P. and KAMPE` DE FERIET, Functions

hypergéométric et hypersphérique, polynome d’Hermite (Gauthier-Villars, Paris) 1926.

[4] DATTOLIG. and TORREA., Thory and Applications of Generalized Bessel Functions (Aracne, Roma) 1996.