Radiat. Phys. Chem. Vol. 50, No. 2, pp. 133-139, 1997
Pergamon PII: S0969-806X(96)00191-0
© 1997ElsevierScienceLtd. All rights reserved Printed in Great Britain 0969-806X/97 $17.00+ 0.00
GENERALIZED HERMITE AND LAGUERRE POLYNOMIALS AND ASSOCIATED BIORTHOGONAL FUNCTIONS
BIORTHOGONAL SETS A N D GENERALIZED POLYNOMIALS G. DATTOLI, A. TORRE and G. M A Z Z A C U R A T I t ENEA, Dipartimento Innovazione, Settore Fisica Applicata, Centro Ricerche di Frascati, C.P. 65 00044 Frascati, Rome, Italy (Received 12 June 1996; revised 29 November 1996; accepted 7 December 1996)
Abstract--43eneralized Hermite functions, with many variables and many indices, provide a set of biorthogonal functions whose importance in applications has been recently recognized. We remark on some of their properties, like those relevant to the Fourier-transform, and exploit them in order to introduce new classes of Laguerre polynomials along with the associated functions, whose properties are studied in detail. © 1997 Elsevier Science Ltd
INTRODUCTION
A number of recent investigations (Dattoli et al., 1995; Dattoli and Torre, 1995a) have been devoted to the theory of generalized Bessel functions and Hermite polynomials. This type of multi-index and multivariable special functions is recurrent in a large body of problems both in pure and applied mathematics (Dodonov et al., 1994). The analysis developed in Refs Dattoli et al. (1995) and Dattoli and Torre (1995a) has led to results which have gone beyond the specific field of special functions and have offered the possibility of speculating on other topics such as, for example, the theory of biorthogonal spaces and on the non-Hermitian realization of Weyl and angular momentum groups. It has, indeed, been shown that within the context of multindex multivariable generalized forms of Hermite polynomials, one can introduce entangled harmonic oscillator biorthogonal functions, with creation annihilation operators exhibiting the peculiar properties of not being adjoint to each other. In this paper we make a step forward in the intriguing land of generalized special functions (G.S.F.) by introducing new forms of Laguerre polynomials (L,P.) and of Laguerre functions with many indices and many variables. We will prove that this class of L.P. and the associated functions can be defined in four distinct ways, which provide two sets of biorthogonal functions. The paper is organized as follows. In Section 2 we review some properties of the generalized Hermite
polynomials (G.H.P.) and of the relevant functions, with particular attention to those relative to the Fourier transform. In Section 3 we discuss the theory of the G.L.P., whose properties are derived from those of G.H.P. The G.L.F. are introduced too, the relevant theory is sketched and the main aspects are discussed. Finally, Section 4 contains concluding remarks.
GENERALIZED
HERMITE
FUNCTIONS
According to App~l and Kamp~ de Feriet (1926), G.H.P. are defined through the generating function I
~c
n
k
e.rUx_ ,;.ru. = ~ -ntUl - ~U2 H , . , ( x , y ) ,, 0 "-- '
(1)
where u and x are real vectors and h~/is a real 2 x 2 matrix, namely
x =
(x) (u) (: ,u=
Y
Au = ac-
M =
(2)
U2 '
b 2 > 0, a,c > 0
The diagonal elements of .~4 are assumed to be positive to ensure the positive definitions of the associated quadratic forms. The polynomials H,,k(x,y) can be explicitly derived from the Rodriguez formula, easily inferred from equation (1) as I
H , , k ( x , y ) = ( - - 1 ) " + k e 2 xT~x t3"+k e t-xr~x ax.Oyk 2
tENEA student. 133
(3)
134
G. Dattoli
The recurrence relations obeyed by H.~ can be written in the form (O, -= Ola~,x = (x,y))
e t al.
= (A_)'
:
ix
B_ =
~is-s.~(=) )
:"'\kU.,_,(x))
M-'O,,
-
(y A+
=a=+
,,.
(4)
and it can be checked that they satisfy the relations and
("~+,,,(')3 ~,:""~-,,,(')~ H.,k + ,(x)) d="~,(')3_ "-\yH,,,,(x)) ""tkkH.,~_ ,(x)) (5) which can be exploited to derive the differential equations obeyed by H~,,, namely (Dattoli and Torre, 1995a; App61 and Kamp6 de Feriet, 1926) (a: -
(aJaD)
(x~a, - a r M - ' O D H , , ( x ) = (n + k ) H , , , ( x )
(6)
The associated polynomials G,., are defined in a similar way and the relevant Rodriguez formula writes
+,
~+t~.,°)-
0
.~)t~.:+,)
~- ='.1 :
o2)
~)t='..-,)
and analogously for the B+ acting on G,.=. The properties under Fourier transform (F.T.) of the G.H.F. have been touched on in Refs Dattoli and Torre (1995a, 1995b), here we consider the problem more carefully. By using the usual procedure and denoting the F.T. operation by F and its inverse F - ~, we have
1
Iw O~"+#< G,~,(x) = ( - 1~. , +,e i-..rM~ - ~ * e - ~ ,.r,it'.,
(7) F(dg,,,,) ( P , ~ ) ~ l
w(0
i
x
It can be shown that the G.,, satisfy the same differential equations as the H.,k. The generalized Hermite functions are introduced as follows (Dattoli and Torre, 1995a) •~..k(X) = ~
f~.,,(X)
-
4~//-~M - -
f~ Jt°,,,.(x,Y)
t ~ --~r~¢~ ,----- H ,,,~x,e 4
(8)
'
e- 7cp=+¢"dx dy~-~-.,m(p,~)
" ,.'.'-4re J,~2 n'm'v~Y~2
,In:k;
(~ ~
,ao -
(°0
a~ ,e _: ( p , O )
G.,k(X)e--4"rM=
F-'(F(~.,..))
They have been proven to satisfy the biorthogonal relation .'ff .,,(x )~., ,,(x ) d x =
6.,.,6,x
(9)
= .J~t'~.,m
(14)
F(a~a~.,.,) (~r) = ~ie F .t Ut~.,~) (e) .
F(x..Ug.,.,) (a) = 2idoF(a~'..=) (a)
which allow us to turn equation (10) in the Fourier space as
and the differential equation
F
7(.@.,.(<~)'~ -~:~,-,~o-l+ ~1 erTM_ ,<'/t~,m(~))
L - arM-'a,- 1 + ~ x M=jt ' ~°.;;;(x)) =(.
d(
The ordinary F.T. operational rules can be specialized as
x/nSkS 1 , _
y
(13)
+
.(..'e.,.(x)'~
mJt~.,.(x))
which is the entangled generalization of the ordinary harmonic oscillator equation. The creation annihilation operators for aff.,~ and ff.~. are defined below .~+ = ~l)l'4rx - 0=,.~_ = )~,/-'c3=+ I x
.(~-.,.(a)~
(10)
(ll)
=
(n + m)t~.,.,(o) )
(15)
with #..m denoting the F.T. of the associated function G..m i.e.
~...,(~)
= F(f¢...,)(~)
The explicit form of 3~'.,., can be obtained by Fourier transforming both sides of the identity
Generalized Hermite and Laguerre polynomials I
__
.,.,iF'= o him!
135
I
e.~M. . . .~ ~*~. . . .~ ~M~ (16)
is ensured by the identity
which yields
2dx~(x')6(
n nl
]
or what is the same (Dattoli and Torre, 1995a,b)
The generalized F.T. yields for ~.,. following functions .~
d~-':
I
--
i)"+"
G.,.,(a)e- ;"M- '<' (19)
It is easy to prove that the biorthogonality is preserved in the Fourier space. We can write the creation annihilation operator of Refs (Dattoli et al., 1995; Dattoli and Torre, 1995a) F(.4 ) = / l + = i5'& F(~I_)=A
(
- i) "+°
and c~.,~ the
"d ~7
x H~,~(k4-'{)~)exp(- I c r Q M - ' Q . )
The same procedure provides for ~.,. the following expression (
(25)
(18)
x/AMW2~Z
--
4
1
( iin+m 4x~.~ H . , , , , ( 7 1 4 - ' a ) e - - 4
~n,m(a)
=
[
Ui~'---~..,.(a ) -- 4Ax//A~ eT"rJ#'ei°r'e-74"rJl-'¢ (17)
,~-n,m(O')=
½Q x - ~l 0 x')
= 2h4-'a+iO:
(20)
(26)
satisfying the following partial differential equation [ l t 4 ~ Q M - '{7~ - 1 - O,~MQ-'~,]
( °'~"(a)'~ e , L ( . ) ) = (n + m) {~-~,,.(a)'~ (27) The creation annihilation operators in Q-Fourier space are derived from equation (20) and equation (21) by replacing a with Qa. Before closing the section we want to stress that integral representations analogous to that holding for the ordinary H.P. can be obtained for ./g.,~ or G.,., too. As an example we provide the identity
and F(B+)=B+ = -(A_)*=
~M-i ^ 'a-i0.
(21) × e7 "T~x e- ~x'Txq-~X'x'"y' ~e~x~dx' dy' OR2
i F(B_ ) = B_ = - (A + )* = is14¢9,+ ~ a The above operators acting on a~-.,=and ~.,~ behaves as A ± and B± in equation (11). We can generalize the definition of F.T. by using the prescription Ft0)(f)(o) =
d x f ( x )e - -2.~¢x: 2
(22)
The obtained results form the basis of the forecoming discussion. GENERALIZED LAGUERRE POLYNOMIALS
The ordinary L.P. are defined in terms of H.P. according to series (Lebedev, 1972) x 2 + y2
Lp( ~ (F{O))-
I(g)(/)
dag(a)e-2"TO+
=
(28)
)=
(-
2plf
~, H2~(x)H2p_2~(y) n!(p - n)! n-O
(29)
2
where Q is a matrix with the same restrictions of.g/. The operational rules (14) are generalized as i
FtO-'(axf)(a) = ~ OoF(O~(f)(tr)
(23)
The obvious generalization of equation (29) to the case of G.H.P. is the following t~,(z) ' -
(-
1)~÷~ " ~ H ~ , ~ ( x ) n ~ _ ~ , ~ _ ~ , t y ) -2p-'~q ~ok~o n!(p -- n)!k!(q - k)!
(30)
Z = (Xl X2 Yl y2),X ~" (Xl X2),y = (Yl Y2)
FCO)(xf)(a) = 2 @ - '~.F(0'(f)(¢)
and the invertibility of the F t0) transform, namely
The differential equation satisfied by Lp,q can be inferred from equation (6) in the form
(F '~) - '(F~0)(f))(x) = f ( x )
(zrc3, - d r c - 'O,)L,,q(Z) = 2(p + q)L,,q(Z)
(24)
(31)
136
G. Dattoli et al. I
where d" is the block matrix given below
r(r...@) - ~ r...(d'-'~)e- ;ee-,, x/2n.4st
The recurrence relations can be derived for Lp~ using equation (4) and equation (5), thus obtaining
It is clear that the polynomials (30) and (35) are not the only realizations of G.L.P. It is indeed possible to introduce the two further 'hybrid' forms
O. L,,q(Z) = (ax, + bx2)L.,q(z) p
(
+ -~(cdx, - bdQL.+,.q(Z)
(33)
q
H2.,z~(x )G2e_ ~,h _ z~(Y)
°-0,=0 n!(p - n)!k!(q - k)! (40)
'
or in a more synthetic form (Ol -- C-'z)nLp'q(Z) = (C-- IOg)nLp + I~n,,+ fin,3,q +fi
'n,4(z)'
n = 1,2,3,4
where (a). denotes the n th components of the a vector and 6.,~ is the Kronecker symbol. Along with Lp,q we can introduce the polynomials
which satisfy equation (38) and the further relations
[--(~
Tpq(Z)
( 2ply~q+q.~=o,~o G2"2*(x)G2e--~h'-z~(n)!k!(q Y)n!(p - k)!
.=0k:0 n!(p - n)!k!(q - k)!
'
(34)
~)g'~-C--''z}WP+'a.l--'n."q-b',.'--'nA (~)
(35)
= [o._ Oo)N., ,4l which are easily shown to satisfy the same differential equation as the L.,, and the recurrence relations (O,).T.,~(z) = ( -
z + C-'O,).T._
~°.,
_~..,~_~..~_,..,(z), n = 1,2,3,4
Again in analogy to the ordinary case, we introduce the generalized Laguerre functions (G.L.F.) defined below
LPp,q(Z) =
Lp,q(z)e-
O)z}Sp,q(Z),
T~,,(z)e- ~,.e:
(36)
whose biorthogonality, i.e.
lYp,q(Z) =
s.Az =
IRZep,q(z)Tp,¢(z) dz = ~p,~,6~,¢
l.¥p,q(~.)e-
l ~../~.,~(z)'~
( _ dry?-,~ _ 2 + -~ Z t.z) t Tp.q(Z) )
.( ze.~(z)'~
= 2(p + q)t Tp.~(Z))
aM S . A z ) e - -,~'~"- -..s, ,.,
It should be understood that it is not strictly necessary that the G.L.P. be defined as a bilinear sum of G.H.P. provided by identical generating functions. We could, for example, define Lp,, such that the two polynomials on the r.h.s, of equation (30) belong to generating functions with matrices h~/~and ~/2. In this case, we can obtain the relevant characteristic equations which can be derived by defining ~" in equation (38) and equation (34) with
~" = (~0-/0' O2)
x/2naM
(42)
(38)
The properties of equation (36) under F.T. are also fairly interesting, we infer indeed, directly from equation (13) and equation (18) : 1
-
(37)
is a consequence of equation (9). The differential equation satisfied by L.ep,qand Tp,q is obtained from equation (15) and reads
F(,~Tp.q) (~) =
n:1,2,3,4
The polynomials Wp,q and Sp,q can be exploited to define a new set of biorthogonal functions, namely
-4~
,
Tp>.(z) =
:[0,-(:
I
- i~)e_ ~TrC-i~ Lp.q(C-
(39)
(43)
So far we have presented only a few of the possible criteria useful to introduce generalized forms of Laguerre polynomials, other examples will be presented in the concluding section.
Generalized Hermite and Laguerre polynomials CONCLUDING REMARKS
As already remarked the consideration of the previous section do not exhaust all the possible way of generalizing Laguerre polynomials. To give further examples we can quote two variable one index generalizations based, e.g. on the Gould-Hopper extension of the ordinary H.P. (Gould and Hopper, 1962). An alternative definition of G.L.P. can be obtained by exploiting the Gould-Hopper type polynomials H~~ defined by the generating function
137
From equation (50) it follows that equation (51) verifies the differential identity Ixd~ + x ~ --
+ y2
1
m m- [
m' 1
(8x + x)" + y~?,.
(Oy + y)m'l.~"'m')(x,y ) =
=
(52)
2p.~a~m."')(x,y)
(a)
£ H~'lCx~ e...... = ~ n=0/'/! n ~
(44)
:
:
satisfying the recurrences (Dattoli et al., 1995)
d H~,,)(x) = m n H ~ ~(x) dx H~+ ,(x) = mH(:)(x) - ~
l( x;
(45)
n~')(x)
and the differential equation
+ mm - Ix -~x +mm
H~'n)(x)= 0 (46)
-
(b)
We define a new class of L.P. involving the H~m) polynomials as follows (m)
L~""')(x,y)oc( -- IF ~
(m')
H2. (x)H~,_ 2.(Y) n!(p - n)!
(47)
n=0
From equation (46) it follows that the G.L.P. equation (47) satisfies the differential equation
-~x + y -~y + 2p L(pm')(x,y) = 0 (48)
(c)
Moreover by exploiting the functions introduced in (Dattoli et al., 1995) ¢(.")(x) =
1
H~,.)(x)e_ l_,a 2
(49)
and satisfying the equation
[X~x-.b x 2 - 1 (m~ x' - '
+ X)m]~m)(x) = nq~m)(x) (50)
w e can suggest the further generalization of Laguerre
functions as
Fig. 1. Three-dimensional plots of L~.0(x, x2, YL, Y2). (a) L~,o(x~, x2, O, 0); (b)Lt.0(xt, 0, 0, Y2); (e) Lt.o(Xt, O, y,, 0).
.Lt'(y"')(x,y)oc(- l y ' ~ ~b~')(x)¢t~")-2"(Y) (51) ,,o n!(p n)! -
138
G. Dattoli et al. satisfy the eigenvalue equation
(a)
=2(p+
1 )..~.(x2 + y2) (54)
which can be viewed as the stationary Schr6dinger equations of spherically coupled harmonic oscillator, ""
5
""
"I
5
"J
(a)
(b)
(b)
(c)
(c)
Fig. 2. Three-dimensional plots of L2.0(x~, x2, yt, Y2). (a) L2,o(xl, x2, O, 0); (b) L2.o(X, O, O, Y2); (c) L2,o(Xl, O, Yl, 0).
-5
However, the search for this type of generalizations might be spoiled of any real interest if not inspired by a global strategy or by specific applications. To better appreciate the usefulness of the so far discussed generalization we remind that the ordinary Laguerre-Gauss functions
Fig. 3. Three-dimensional plots of L2j(xt, x2, y~, y2), (a) L2j(Xl, X2, O, 0); (b) L2,1(xl, O, O, Y2); (c) L2,t(xI, O, Yl, 0).
1 ;I (.~ + -,2) (53) "~P(x= + Y:) = - ~ n L~(x2 + y2)e -
Generalized Hermite and Laguerre polynomials useful to treat problems like the evolution of coherent states of the binomial type (Dattoli et al., 1987; Stoler et al., 1985), which occur in the physics of free-electron laser. Equation (38) is the four variables, two indices extension of equation (54) and its physical interpretation may be again that of a Schr6dinger equation describing coupled-entangled harmonic oscillators. The remarkable feature is that this type of equation is satisfied by four distinct G.L.P. forms which provide two sets of biorthogonal functions, whose behaviour is given in Figs 1-3.
REFERENCES
App61, P. and Kamp~ de Feriet, J. (1926) Fonction
Hyperg$om$triques et Hypershpbriques, d'Hermite. Gauthier-Villars, Paris.
139 Polynome
Dattoli, G., Gallardo, J., Maino, G. and Torre, A. (1987) JOSA !i4, 185. Dattoli, G., Lorenzutta, S., Maino, G. and Torre, A. (1995) Ann. Numer. Math. 2, 211. Dattoli, G. and Torre, A. (1995a) J. Math. Phys. 36, 1636. Dattoli, G. and Torte, A. (1995b) Riv. Nuovo Cimento ll0B, 1197. Dattoli G., Lorenzutta S., Maino G., Torre A. and Cesarano C. (1996) J. Math. Anal. Appl. 203, 597. Dodonov, V. V., Man'Ko, O. V. and Man'Ko, V. J. (1994) Phys. Rev. A50, 813. Gould, H. W. and Hopper, A. T. (1962) Duke Math. J. 29, 51. Lebedev N. N. (1972) Special Functions and Their Applications. Dover, New York. Stoler, D., Saleh, B. E. A. and Teich, M. (1985) Opt. Acta 32, 345.