ENTE PER LE NUOVE TECNOLOGIE, L’ENE13GIA E UAMBIENTE Dipartimento Innovazione FROM HERMITETO HUMBERTPOLYNOMIALS GIUSEPPE DAll_OLl ENEA - Dipartimento Innovazione Centro Ricerche Frascati, Roma SILVERIA LORENZUTTA ENEA - Dipartimento Innovazione Centro Ricerche “Ezio Clementel”, Bologna CLEMENTE CESARANO ENEA STUDENT RT/lNN/99/22 This report has been prepared and distributed by: Servizio Edizioni Scientifiche - EN EA Centro 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. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. SUMMARY New integral representations of Chebyshev and Gegenbauer polynomials in terms of Hermite polynomials are introduced. It is shown that most of the properties of these classes of polynomials can be deduced in a fairly straighflorward way from this representation, which provides a unifiing framework for a large body of polynomial families, including generalized forms of the Humbert and Bessel type, which are a natural consequence of the point of view developed in this papez (CHEBYSHEV, GEGENBAUER, HERMITE, BESSEL POLYNOMIALS, INTEGRAL REPRESENTATION) RIASSUNTO Si introducono nuove rappresentazioni integrali dei polinomi di Chebyshev e Gegenbauer. Si dimostra che molte delle propriet~ di queste classi di polinomi possono essere dedotte da tale tipo di rappresentazione, che costituisce un elemento unificante per un Ia[go spettro di famiglie polinomiali, che includono forme generalizzate di polinoml di Humbert e Bessel. INDICE 1- INTRODUCTION .......................................................................................................p. 7 2- HERMITE AND GEGENBAUER POLYNOMIALS ................................................p. 9 3- HERMITE AND UMBERT POLYNOMIALS .......................................................... p. 13 4- CONCLUDING REMARKS .....................................................................................p. 15 REFERENCES .......................................................+.......................................................p. 18 FROM HERMITE TO HUMBERT POLYNOMIALS 1- INTRODUCTION It is well known that Chebyshev polynomials of the second kind are defined by [1] {ni~l (_l)k(n _ k)!(2x)n-2k Un(x) = ~ (1) k!(n – 2k)! k=O and that two variable Hermite polynomials reads [2] [n’21 ykxn-2k I&(x,y) = n! ~ (2) k!(n–2k)! “ k=O 13yrecalling that m n!= Je–ttndt, (3) o we can state the following integral representation for the second kind Chebyshev polynomials cm Un(x) = ~ ~e-ttnHn 2x, –~ dt (4) o () Along with (4) we introduce the further representations I@ e–ttn–l Tn (X)= Hn 2x, –~ dt , 2(n–l)! Jo () t (5) 2W e–ttn+l Wn(x) = Hn 2x, –~ dt . (n+l)! oJ () t I I 8 The polynomials Tn(x) are first kind Chebyshev polynomials,[ 1] while the nature and the role - of the Wn(x), which for the moment will be indicated as third kind Chebyshev, will be specified in the following. Just to give a preliminary idea of the usefulness of the previous representation, we note that the following well known properties of the two variable Hermite polynomials a ~%(x,Y) = n%-l(x,y) (6) Hn+l(x,y) = Hx+2y: Hn(x, y) , allow to conclude that the Un(x), Tn(x) and Wn(x) are linked by the relations -& Un(x) = nWn_~(x) , (7) ~Tn(x)= nun-l(x) , and un+~ (x) = Xwn (x) – --&Wn.l(x) , (8) Tn+l(x) = Xun(x) – un_l(x) . Furthermore we can obtain the generating functions of the above families of Chebyshev polynomials, directly from that of the Hermite polynomials ~ ~EIn(x,y) = exk+yc’. (9) n! n=O By multiplying indeed both sides of Eq. (4) by ~n/n!, by summing up over n, by exploiting (9) and then by integrating overt, we end up with ~g”u.(x)=1_2E;+g2 . (lo) n=() By using the same procedure and the second of Eqs (6) we get (-l<x<l, 1~1<1) 9 (11) ~ (Il+l)(rl+zpwn+,(x) = ‘(x-~),, . n=o (Wx+~ ) The use of integral representations relating Chebyshev and Hermite polynomials is a fairly important tool of analysis allowing the derivation of a wealth of relations (others will be discussed in the concluding section) between first, second and third kind Chebyshev polynomials. This paper is devoted to a first attempt towards this direction. We will show that Gegenbauer polynomials can be framed within such a framework, which offers interesting criteria of generalizations. It will be indeed shown that, by combining the wealth of Hermite polynomial forms and the flexibility of the proposed representations, we can develop a systematic procedure of generalization by including in a natural way the Humbert, [3] Gould [4] and Bessel [5] polynomials. 2- HERMITE AND GEGENBAUER POLYNOMIALS Integral transforms relating Chebyshev and Hermite polynomials are not new, in Ref. [6] It has been shown that: co Un (x)= ~ ~ e-sHn (2xs,–s)ds (12) -o we can introduce a generalization of Un(x) by defining the quantity cm Un(x,y;a)=~je ‘asHn(2xs,–ys)ds (13) .0 The use of the identity ~Hn(x,y) = n(n – l) Hn_2(x,y) (14) ay allows to conclude that 10 a a —Un(x, y;cx)= #Jn_2(x, y;a) , ay (15) a ~un(x, y;(x) = –2&n-l(x, Y;(x) , which can be combined to give (32 a2 —Un(x,y;a) = 4 —Un(x,y;a) . (16) ax2 actay This last identity and the fact that Un(x,o;a).q (17) a allows to define the generalized Chebyshev polynomials as ;Dil$ Q& . Un(x, y;a) = e a n+l (18) ‘n/2] (-y)s(n - s)!(2x)n-2s = z n+l–s S=o s!(n–2s)!~ where we have denoted by 6;1 the inverse of the derivative operator, the rules concerning the use of this operator will be discussed in the concluding sectibn. The polynomials Un(X,y;U) can be specified by a further integral representation generalizing that given in Eq. (4), we have indeed w Un(x,y;a)=~je ‘attnHn 2X,–Z t (19) () t .0 which suggests the introduction of the polynomials Tn(x,y;cx), Wn(x,y;a) (see Eq.(5)) linked by $Un(x,y;oo = - ~(n+l)wn(x,y;a) (20) a ~Tn(x, y;cx) = –~un(x,y;~) 11 Chebyshev polynomials are particular cases of the Gegenbauer polynomials specified by the - serie 1 1 ‘n’21(–l)k(2x) ‘-2kr(n – k +v) C\P)(x) = — z (21) ‘(~) k=~ k!(n – 2k)! ‘ by using the integral representation of the Euler functionl co r(v) = J e-ttv-ldt (22) o and the same arguments exploited for the Chebyshev case, we can introduce the polynomials 1 [n ’21 (–y)sr(n + p – S)(2X)*-2S = c~~)(x,y;a) = — x ‘(~) s=o s!(n – 2s)!cxn+p-s (23) The relevant generating function is readily obtained by means of the same procedure leading to Eqs. (10, 11) thus getting indeed (v#O) (24) The integral representation (23) is very flexible and can be exploited for different purposes. It is e.g. evident that (-,)m ~l-Jn:: (x,y;ct)= m! C\m+l)(x, y;cx) , (25) and ~(-&)m-’c\m)(X,~;U)=c~)(x,y;U+<)=U*(x,~;a+~). (26) m=l The recurrences n+l (p) —C*+* (x, y;(x) = XC$+Q(X, y;a) – ycg-l)(x, y;a,) (27) 2~ 12 and %$~)(x, y;ct) = –p c@+l)(x,y;(X)~_2 . (28) ay are just a consequence pf Eq. (23) and of the identities (6) and (14). The use of the operational relation m Hn+m(X, Y) = ()x + 2Y& Hn(x,Y) (29) has been used to generalize the Rainville identity [7] m in Xt+yt2Hl(x + 2yt>Y) “ x ~Hn+l(x, y)=e (30) n=o “ The previous relations allow the derivation of a wealth of generating functions for the C$w)(x, y;ct) polynomials, we find indeed that (p>O) i (31) 1= J –r(~) ~ e-att’-1’m(2x-2:#e-t(a-2x’ can be rewritten as (n+‘)! gn~$~)m~x, y;uj = 1 m C(V)(X - YE>F(x>Y;GQY;l) (32) z n!m! n=0 [F(x,y;cx>QIN+mm where F(x, y;cx,~) = u - 2x& + Y<z . (33) In this section we have provided further elements proving the usefulness of the link between Hermite and Gegenbauer polynomials, in the following we will show that we can go much further using the wealth of Hermite forms which will allow us to get further interesting generalizations. 13 3- HERMITE AND UMBERT POLYNOMIALS The Kamp6 de F4ri6t Hermite polynomials of order m are specified by 2] the series definition ‘n’m] yrxn-m H\m)(x>y) = ‘! ‘=0 (34) x r!(n – mr)! and by the generating function (35) we can therefore suggest the further generalization of the second kind Chebyshev polynomials by introducing the polynomial cc (m)”n(x’y;~) = ~ Jdte-afnH~mJ (36) “o (mx+) whose generating functions can be easily obtained as (37) The generalized Gegenbauer polynomials can be found using an analogous procedure, thus finding (p>O) (38) with the generating function (v#O) i &’m)c!iv)(x,y;U) = (39) n=() [cX-mx:+Yqp “ The use of the identities 14 “~’(xy)=[x+my:~:JH’m)(x (40) a (m) —Hn (x,y) = n(n – 1)...(n – m + l)H$~~(x, y) ay yields the recurrences n+l (rn)c$/)l(x, Y;U) = x (rn)c~@+l)(x, y;~) – Y (m) c@+l)n–m+l (x,’;(x)) –(m~ )( )( (41) a C$W)(X, Y;U)= ‘~ (m) c@+l+x,n–m ‘;(X) ~ (‘m) )( ) This last class of polynomials can be identified as the Humbert polynomials introduced by Humbert in Ref. 3 within a different context. A systematic effort of generalization of the Gegenbauer polynomials was also undertaken by Gould [4].