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
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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 (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~$~)m~x, y;uj = 1 m C(V)(X - YE>F(x>Y;GQY;l) (32) z n!m! n=0 [F(x,y;cx>QIN+mm where F(x, y;cx,~) = u - 2x& + Y In this section we have provided further elements proving the usefulness of the link between Hermite and Gegenbauer polynomials, in the following we will show that we can go much further using the wealth of Hermite forms which will allow us to get further interesting generalizations. 13 3- HERMITE AND UMBERT POLYNOMIALS The Kamp6 de F4ri6t Hermite polynomials of order m are specified by 2] the series definition ‘n’m] yrxn-m H\m)(x>y) = ‘! ‘=0 (34) x r!(n – mr)! and by the generating function (35) we can therefore suggest the further generalization of the second kind Chebyshev polynomials by introducing the polynomial cc (m)”n(x’y;~) = ~ Jdte-afnH~mJ (36) “o (mx+) whose generating functions can be easily obtained as (37) The generalized Gegenbauer polynomials can be found using an analogous procedure, thus finding (p>O) (38) with the generating function (v#O) i &’m)c!iv)(x,y;U) = (39) n=() [cX-mx:+Yqp “ The use of the identities 14 “~’(xy)=[x+my:~:JH’m)(x (40) a (m) —Hn (x,y) = n(n – 1)...(n – m + l)H$~~(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]. The strategy of generalization we are developing is complementary to that of Gould and benefits from the variety of existing Herrnite polynomials. To give a further example we note that the polynomial [n/ 3] ~rH n–sr(x>Y) Hn(x,y,z) = n! ~ (42) r=() r!(n – 3r) ! can be exploited to define (p>O) lm &e-ctttn+~–lH C$)(’, Y, Z;(X) = r(~)n! o J n (43) and the generating function (44) can be exploited to prove that (v#O) (45) 15 This function too has interesting properties which will be more deeply investigated in a forthcoming paper. In this section we have outlined how a systematic effort to generalize Gegenbauer polynomials can be achieved in a fairly straightforward way starting from the properties of Hermite polynomials in the forthcoming section we will discuss further examples and discuss future direction of the present line of research. 4- CONCLUDING REMARKS We have left open some points in the previous sections which we will try to clarify in this concluding part. We have introduced the polynomials Wn(x) which have been indicated as third kind Chebyshev polynomials, it is clear that in terms of Gegenbauer polynomials they can be recognized as Wn(x)= —2 C$2)(X) (46) n+l Furthermore by recalling that 1 Tn (x)= cos(n Cos-l (x)) , (47) sin[(n + I)cos–l(x)] U*(X) = m’ we also find –x(n +1) sin[(n + 2)cos–l(x)] (n+ l)Wn(x) = ~_ ~Z n‘2 cos[(n + l)cos-l(x)]. (48) m - J~- R isalso worth nothing that Un(x, y;u) = ;l(yy’2un[jL); (49) 16 which can be exploited to get Un(x,y;cx) and Tn(x,y;u) in terms of circular functions. We want - to spend few words more on Chebyshev polynomials to note that from Eqs. (47) we get (-1-<1) (50) and (51) and Generating functions involving squares of Chebyshev polynomials can be obtained too, thus finding e.g. ~ ~[Tn(x)]2 = ~e<[l +exp(2~(x2 -l))cos(2x~~)] . (53) .=() - These last relations can be used as a useful complement of the generating functions given in Eqs. (10, 11). The use of the method of the integral transform can be usefully extended to other families of polynomials, a noticeable example is provided by the Bessel polynomials [5], defined by n (n+k)! x k y.(x)= k~ok!(n– k)! ()~ (54) or, equivalently, by the integral representation n Yn(x)=~~e-t t+< dt. (55) 2 .0‘[l This last identity suggests the following generalized forms of Bessel polynomials (cx=real) 17 (56) with the interesting properties a ~Yrl(x,w;~) = +Yn-l(x,w;a+2) +Yn(x,w;~)=Y~-l(x,w;a+l) (57) (n+ I)yn+l(x,w;a) = Wyn(x,w;a + l)++yn(Lw;CX+2) and (58) Regarding the use of the operator D~l we note that it acts on functions of the type f(ct)=orrn so that we find ~;s~–m+l (m – S)!~_rn+l_S = (59) m! We have not specified the lower limit of integration which is tacitly assumed to be infinite. The results we have presented in this paper have been shown to be a versatile tool of analysis to study the properties of classical and generalized polynomials. In a forthcoming investigation we will benefit from the present results to go more deeply into the properties and the applications of the discussed polynomials. 18 REFERENCES [1] P.J. Davis, Interpolation and Approximation, Dover, New York (1975) [2] G. Dattoli and A. Terre “Theory and applications of generalized Bessel functions” ARACNE Rome (1996) [3] P. Humbert “Sur une g~nkralisation del’equation de Laplace”, J. De Mathkmatiques Pures ed Appliqu&es (9), vol. 39 pp. 145-159 (1921) [4] H.W. Goud “Inverse series relations and other expansions involving Humbert polynomials” Duke Math. J. pp. 697-711 (1965) [5] H.L. Krall and O. Frink “A new class of orthogonal polynomials: the Bessel polynomials” Trans. Amer. Math. Sot. 65, pp. 100-105 (1948) [6] G. Dattoli, S. Lorenzutta, A. Terre, “The generating function methods and evaluation of inegrals involving combinations of special functions” Quaderni del Gruppo Nazionale per l’Informatica-Matematica del CNR Roma (1997), n. 8 [7] E.D. Rainville, Special Functions, MacMillan, New York (1960)