ENTE PER LE NUOVE TECNOLOGIE, L'ENERGIA E L'AMBIENTE IT0100533

Serie Innovazione

A NOTE ON OPERATIONAL RULES FOR HERMITE AND LAGUERRE POLYNOMIALS

GIUSEPPE DATTOLI ENEA - Divisione Fisica Applicata Centro Ricerche Frascati, Roma

SILVERIA LORENZUTTA ENEA - Divisione Fisica Applicata Centro Ricerche "Ezio Clementel", Bologna

DARIO SACCHETTI Universita degli Studi di Roma, "La Sapienza", Rome (Italy) Dip. di Statistica, Probability e Statistiche Applicate,

RT/INN/2000/23 32 This report has been prepared and distributed by: Servizio Edizioni Scientifiche - ENEA Centra Ricerche Frascati, C.P. 65 - 00044 Frascati, Rome, Italy

The technical and scientific contents of these reports express the opinion of the authors but not necessarily those of ENEA. A NOTE ON OPERATIONAL RULES FOR HERMITE AND LAGUERRE POLYNOMIALS

Riassunto Si presentano identita operazionali utili in problemi che coinvolgano forme ordinarie e generalizzate di polinomi di Hermite e Laguerre

Abstract Operational identities, useful in any problem involving ordinary or generalized forms of Hermite and Laguerre polynomials are presented.

Key words: operational rules, Hermite, Laguerre, Kampe de Feriet, Bell, quantum optics, combinatorics INDEX

1. INTRODUCTION 7

2. HERMITE POLYNOMIALS OF MANY VARIABLES AND ONE INDEX 7

3. LAGUERRE POLYNOMIALS WITH MANY VARIABLES AND ONE INDEX 12

4. MULTI-INDEX HERMITE POLYNOMIALS 14

ACKNOWLEDGEMENTS 17

REFERENCES 18 A NOTE ON OPERATIONAL RULES FOR HERMITE AND LAGUERRE POLYNOMIALS

1 INTRODUCTION

Increasing interest has grown during the last years around operational methods and special functions, within the context of problems concerning quantum optics [1] and combinatorics [2]. Most of the interest is relevant to operational identities associated with ordinary and generalized forms of Hermite and Laguerre polynomials. Some of these identities, discovered since long time are not widespread known as they sould be and have been the subject of recent investigations and re-discoveries. We consider therefore useful to present a brief, but organic, note on a set of "practical" identities, which can significantly simplify calculations involving families of special polynomials.

2 HERMITE POLYNOMIALS OF MANY VARIABLES AND ONE INDEX

The Kampe de Feriet version of Hermite polynomials writes [3]

Hn(x,y) = n! r=0 v

n Hn(x,O) = x

Their properties are summarized below n=0

^-Hn(x,y) = n(n-l)Hn_2(x,y), (2) 9y

—- H (x,y) = nH _!(x,y) ox n n

The recurrences can be combined to get

a a2 — Hn(x,y) = -THn(x,y) (3) ay dx This last identity and the second of eq. (1) can be exploited to derive the operational definition

y 5 2 n Hn(x,y) = e * (x ) . (4)

The ordinary forms of Hermite polynomials are particular cases of Hn(x,y) and indeed we have

[n/2] r(2x)n-2r " " ±^ r!(n-2r)!

a straighforward consequence of identity (4) ensures that nnw = e (6)

23x2 n Hen(x) = e (x ) .

The first of eqs. (6) traces back to the second decade of the last century [4] but does not appear very well known (see further references and comments by A. Wiinsche in Ref. [1]).

The identity

3x2 Hn(x,y + z) = e Hn(x,y) , (7) is a further consequence of eq. (4), and provides as important by product, the inverse of the identity (4), namely (see refs [1,5])

n ax2 x =e Hn(x,y) (8) and

2 I a 43x n e Hn(x) = (2x) , (9)

23x2 n e Hen(x) = x .

An interesting identity summarizing all those presented so far is

a2d-c2b d2 (ac)2 5x22 Hn(ax,by) . (10)

By specifying eq. (10) to the ordinary case we find 10

(11)

A further important set of polynomials which plays a significant role within the context of multi-photon processes is provided by the following extension of the Kampe de Feret family, namely

[ii/ IIIJ ( m) yrxn-mr H n (x .y)-n r=0 ridi-mr)!

( m) n H n (x ,0) = x

The superscript m is usually neglected for m=2.

This family of poplynomials satisfies the recurrences

(m) — HCx A H (x v\ ay (n-m)! (13) which leads to the identiy

m Hn(x,y) (14) By dx it is therefore evident that the results relevant to the case m=2, can be generalized by replacing —~- with 9x2z 3x m

A more interesting aspect of the problem is associated with the multivariable extension of the previous families of polynomials. We introduce the p-variable Bell type polynomials [6]

r!(n_pr)! 11

with generating function

2 S £ 44 '-'P)(x1,...,xp) = e^.^ (16) i n=0 and recurrences

n JLH(2,...,p)f , ' H(2,-,P)( F dxs (n-s)!

It can be checked that the following identities hold

( 2 ) 2 Sd H n '-'P (x1,...,xp) = e " *k4) (18) and

ax, (H(2,...

x ( 2 p) ( 2 p) e ' H n '-' (x1,...,Xp) = H n '-' (x1,...,Xp+p,...,xp), l

dx ( 2 ) ( 2 c ) e >H n '-'P (x1,...,xp) = H n '-'P' ' (x1,...,xp,P), q)p ,

[n/q]xrH(2,...,p)(x ^^x ^ p' q ^1 r!(n-qr)!

Relations of the above nature are particularly important and can be exploited to derive identities involving ordinary polynomial as e.g. 12

( 2 m) m (Hn(x)) = H n ' (2x,-I,2 y),m > 2, (20)

The generalization of eq. (10) to the multivariable case writes

b s VP I s as ] 3

u a A 1 p. V l \ J " iuiA-,PJ/Q v „ v \ /"in c nn ^d2Xj,...,aDxD; ^zi;

3 LAGUERRE POLYNOMIALS WITH MANY VARIABLES AND ONE INDEX

Two variable one index Laguerre polynomials have been introduced in Ref. [7], they read

(22)

n! and are specified by the generating function

xt l"yt, |yt|

— Xn(x,y) = nXn_1(x,y), dy (24)

-T-X—Xn(x,y) = nXn_!(x,y) ax ax

The previous last two equations can be embedded to derive the relation 13

—Xj,(x, (25) ay ox ox

and the consequent operational identity

a a x n A! ^ (26) n! analogous to the Hermite case. The operator

Being the ordinary Laguerre polynomials Ln(x) linked to £n(x,y) by

(27)

we find

(28) n!

It is also easily understood that

(29)

n!

The last identities show that it is possible to obtain for the Laguerre polynomials a set of operational rules completely analogous to the Hermite case, we get therefore 14

bc-ad

£n(cx,dy) = |-| e £n(ax,by) . (30)

It is now worth noting that the Laguerre derivative has the remarkable property

m m+k sm y IT1 (31) m k! 3x dxm k=0 and that Laguerre like polynomials satisfying the identity

(32) are specified by [8]

—r^mr (m) X (x,y) = X n S (n-r)!((mr)!)2 (33) ,mn ((mn)!) we can therefore extend to the Laguerre type polynomials the already developed considerations relevant to the Hermite family. We do not dwell on this aspect of the problem for brevity's sake.

4 MULTI-INDEX HERMITE POLYNOMIALS

Examples of multi-index Hermite polynomials are contained in the original Hermite memory (see ref. [3] and references therein) in this paper we will exploit a class of polynomials introduced in Ref. [9] and provided by

[m,n] ^ [m,n] = min(m,n) .(34) r!(m-r)/(n-r)! 15

This family of polynomials, of crucial importance in problems involving entangled harmonic oscillators [9], are generated by

m n v +z + v ^-^-Hm n(x,y;z,w | T) = e^ ™ ™ (35) (m,n)=0 m! n! and the relevant recurrences can be exploited to derive the operational definition

)2 a2 a2 m e c/x- dz~ o^(x z") = Hm;n(x,y;z,w|x) (36) which can be complemented by its obvious inverse and by the more general relation

Hmn(fx,gy;hz,kw|lx) = f-J (-J AHmn(ax,by;cz,dw|qx) (37) where the operator A is specified by

a2q-f2b 32 c2k-h2d 32 Hac-qftA = V ;rH = W r-+ X A = e (af) dx (ch) dz ^ ^^ ' 3x3z (38)

The above family of polynomials is general enough that the two-index family defined on a quadratic form and originally introduced by Hermite [3] can be defined in terms of (34), thus finding

Hm,n(x,y) = Hm,n(ax + by,--a;bx + cy,--c | -b) (39) and the relevant operational definition, which can be inferred from eq. (36)

2 2 2 1 f a OK a a c—^-2b +a— V (40)

A = ac-b2 >0, a, c>0 . 16

A class of two variable Hermite polynomials of noticeable importance is provided by the so called incomplete family [9,2]

3 -(y—y+w M1(x,z|T) = Hm>n(x,0;z,0|T) = (41) defined through the rule

hm>n(x,z|x) = (42) and linked to the Laguerre polynomials by

xz ), n>m x (43)

•—) , m>n X where U$(x)(= £&\x,\)) are associated Laguerre polynomials, which within the present operational framework can be defined as

_yiLxJL (x)m (44) m!

Before closing the present note it is worth mentioning other two points associated with the Crofton formula and the Burchnall type identities.

The Crofton formula states that [10]

e 3xm [f (x)g(x)] = f x + my- (45)

The operator f x + m —p can be handled in a fairly simple way. In the case in which v ox J f(x)=xn and m=2 we find the generalized Burchnall identity [4] 17

n n (2y)sH _ (x,y)- (46) n s ,s' s=0

while for f(x)=Hn(x,z) we find

s (2y) Hn_s(x,y (47) s=0 dx which for y=-z yields the inverse of the Burchnall identity [5]

s n s (-2y) x ~ s (48) s=0 dx

The extension of the above relations to the multivariable case will not be discussed here for brevity's sake, we note that e.g.

(49) the interested reader can find further comments in Ref. [11].

In this note we have presented a collection of "practical rules" for families of ordinary and generalized polynomials, we believe that their use may provide significant simplification of calculation in computational problems involving these families. The extension of the present formalism to other families (Jacobi, Legendre, Gegenbauer....) will be discussed elsewhere.

ACKNOWLEDGEMENTS

The authors express their sincere appreciation to Dr. A. Wiinsche for a stimulating corrispondence and for informations about his work. REFERENCES

[I] G. Dattoli and A. Torre, Nuovo Cimento B110, 1197 (1995)

A. Wünsche, J. Phys. A32, 3179 (1999)

[2] L. Comtet, "Analyse Combinatoire" ed. by Universitaires de France Saint Germaine

(Paris) 1970

[3] P. Appell, Kampé de Fériét, "Fonctions hypergéométriques et hypersphériques

polynômes d'Hermite" Gautier Villars Paris (1926)

[4] S.L. Straneo, Rend. Ace. Lincei, Roma 8, 575 (1928) see also

J.L. Burchnall, Quart. J. Math. Oxford Ser.(2), 9 (1941)

H.W. Gould and A.T. Hopper, Duke Math. J. 29,51 (1962)

G. Dattoli, S. Lorenzutta, G. Maino and A. Torre, Annals of Num. Math. 2, 211 (1995)

[5] G. Dattoli, A. Torre and S. Lorenzutta, J. Math. Anal, and Appl. 227, 98 (1998)

G. Dattoli, A. Torre and S. Lorenzutta, J. Math. Anal, and Appl. 236, 399 (1999)

[6] E.T. Bell, Ann. of Math. 35, 258 (1934)

[7] G. Dattoli and A. Torre, Ace. Se. di Torino-Atti Se. Fis 132, 1 (1998)

Operational methods for associated Laguerre plynomials have also been youched in A.

Wünsche, J. Phys. A32, 3179 (1999)

[8] G. Dattoli, S. Lorenzutta, A. Mancho and A. Torre, J.C.A.M. 108,209 (1999)

[9] G. Dattoli, S. Lorenzutta, G. Maino and A. Torre, Le Matematiche LII, 179 (1997)

[10] M.W. Crofton, Quart. J. Math. 16, 323 (1879)

[II] G. Dattoli, S. Lorenzutta and A. Torre, Le Matematiche LII, 337 (1997) Editodall'l Unite Comunicazione e Informazione Lungotevere Grande Ammiraglio Thaon di Revel, 76 - 00196 Roma Sito Web http://www.enea.it Stampa Laboratorio Tecnografico - C.R. Frascati Finito di stampare nel mese di marzo 2001