International Mathematical Forum, 1, 006, no. 13, 603-616 MONOMIALITY, ORTHOGONAL AND PSEUDO-ORTHOGONAL POLYNOMIALS G. Dattoli Unità Tecnico Scientifica Tecnologie Fisiche Avanzate ENEA Centro Ricerche Frascati, C.P. 65 Via E. Fermi, 45 00044 Frascati, Roma Italia) e-mail: dattoli@frascati.enea.it B. Germano, M.R. Martinelli, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università degli Studi di Roma La Sapienza Via A. Scarpa, 14 00161 Roma Italia) P.E. Ricci Dipartimento di Matematica Guido Castelnuovo Università degli Studi di Roma La Sapienza P.le A. Moro, 00185 Roma Italia) e-mail: riccip@uniroma1.it Abstract We reconsider some families of orthogonal polynomials, within the framework of the so called monomiality principle. We show that the associated operational formalism allows the framing of the polynomial orthogonality using an algebraic point of view. Within such a framework, we introduce families of pseudo-orthogonal polynomials, namely polynomials, not orthogonal under the ordinary definition, but providing series epansions, which can be obtained from the ordinary series using the monomiality correspondence. Mathematics Subject Classification: 33C45, 44A45, 33C50 Keywords: Monomiality, Orthogonal polynomials, Pseudo-orthogonality.
604 G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci 1 Introduction It has been shown that the two variable Laguerre polynomials can be introduced as see ref. [1]) L n, y) =y D 1 )n 1) where D 1 is the negative derivative operator, defined in such a way that D 1 With this assumption we get from eq. 1) L n, y) = n f) = fξ)dξ. ) ) n 1) s y n s D s = s 0 n 1) s y n s s n s)!s!), 3) obtained on account of the fact that, when the negative derivative operator is acting on unity, we find D n = n. 4) The ordinary Laguerre polynomials L n ) are obtained from eq. 3) by setting y =1. The above definition, based on the negative derivative operator formalism, allows a fairly straightforward derivation of old and new relations involving epansions in terms of Laguerre polynomials. A very natural eample is offered by eq. 4) itself, which, rewritten as yields, according to eqs. 1)-3), n = n = y y D 1 ))n 5) n ) n 1) s y n s L s, y), 6) s i.e. the epansion of an ordinary monomial in terms of Laguerre polynomials. It must be stressed that the derivation of the identity 6) is purely algebraic and does not imply any assumption on the orthogonality of the Laguerre polynomials see e.g. ref. [] for an orthodo derivation). The etension of eq. 6) to non integer values of n is easily obtained too, using the properties of the Newton binomial. We want to emphasize that such an algebraic point of view to the epansion in terms of Laguerre polynomial may be a fairly interesting tool to be further investigated for its usefulness in calculations and for possible etensions of the orthogonality concept it may offer.
Monomiality, orthogonal and pseudo-orthogonal polynomials 605 Operational methods and epansion in Laguerre polynomial series Let us go back to the definition 1) and note that can we use the following operational definition to introduce the Laguerre polynomials L n, y) = ep D 1 ) y n 7) the eponential operator, which acts on any function of y as a shift operator, can be further understood as follows ) ep D 1 1) s ) s = D s 1) s ) s = s! s!) s 8) which can cast in a closed form as follows ) ep D 1 = C 0 ) 9) where we have eploited the 0 th order Tricomi function linked to the ordinary cylindrical Bessel function by C n ) = n/ J n ). 10) According to the previous relations we can also conclude that L n, y) =C 0 ) y n y n = ep D 1 ) L n, y). 11) The last of eq. 11) is just a consequence of the fact that ) ) ep D 1 ep D 1 = 1 1) which does not imply that C 0 D 1 ) C 0 D 1 ) =1. 13) We can use the previous relations to get further series epansion in terms of Laguerre polynomials, without any eplicit use of their orthogonality. We
606 G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci start from the definition of the eponential function in terms of the Taylor epansion t n y n epty) = 14) and note that C 0 ) epy) = epy)c 0 ) 15) thus getting, from eq. 11) and 14) epty)c 0 t) = t n L n, y), 16) which is one of the well known generating functions of the Laguerre polynomials ref. []). Let us note that we can eploit other well known epansions like t n y n = 1, yt < 1, 17) 1 yt to obtain, according to the same procedure as before, see ref. [1]) t n 1 L n, y) = 1 ty D 1) = 1 1 yt ep t ), 18) 1 yt which is the ordinary generating function of Laguerre polynomials ref. []). The following well known epansion can be derived from eq. 18) ep ay) = 1 1+a The use of eq. 15) and of the identity C 0 ) L n y) = L L n, y) = leads to C 0 a) = epay) 1+a ) a n Ln y), a > 1 1+a. 19) n 1) r L r, y) n r)!r!), 0) ) a n L L n, y)). 1) 1+a This last relation can be viewed, on account of eq. 10), as a new form of epansion of Bessel functions, which has been obtained using the polynomials, given in eq. 0), without any eplicit use of orthogonality properties and without knowing whether they belong to any orthogonal set. In the forthcoming section we will see how the above point of view can be etended to other polynomial families and show that the concept of polynomial orthogonality can be framed within a more general algebraic contet.
Monomiality, orthogonal and pseudo-orthogonal polynomials 607 3 The pseudo-orthogonality We have introduced the polynomials LL n, y), which can be eploited to derive other epansions directly inspired by those of ordinary Laguerre, we note indeed that we obtain from eqs. 6), 7) the relation C 0 ) n n y n = ) 1) s C 0 ) L s y) ) s which eventually yields L n, y) = n ) n 1) s LL s, y). 3) s In this case too we have not eploited any concept associated with the orthogonality properties of the above family of polynomials. The ordinary Laguerre polynomials L n y) are bi-orthogonal ref. [3]) to φ n y) = ep y)l n y), 4) we can therefore introduce, along with the polynomials 0), 4), their pseudoorthogonal partners defined as Φ n, y) =C 0 ) φ n y) =φ n y D 1 ) 5) which eplicitly writes Φ n, y) = ep y)a n, y), obtained after using the identity n 1) s T s, y) A n, y) = s!) n s)! ) s s T s, y) = y r s 1) s s C s ) r 6) D s C 0 ) = s C s ). 7) The polynomials 0) and the functions 6) are isospectral to the ordinary Laguerre polynomials and to their bi-orthogonal partners but they are by no means orthogonal, in the strict sense of the word. Notwithstanding they can be eploited to define series epansions, like those given in eq. 1) of algebraic nature and not implying any orthogonality condition. We will say therefore that Φ n, y) and L L n, y) form a pseudo-orthogonal set. This concept will be further discussed in the forthcoming sections.
608 G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci 4 The Hermite polynomials: an algebraic point of view to their series epansion The Hermite polynomials are defined within the framework of the monomiality treatment by means of the operational rule ref. [1]) ) [ H n, y) = ep y n n] n r y r = r!n r)! 8) These polynomials belong to the Hermite-Kampé de Fériet family see [4]) and reduce to the ordinary Hermite for H n, 1) = H n ) H n, 1 ) = He n ). 9) It is evident from eq. 8) that we can invert the above definition to get n = ep ) y [ n] H n, y) = y) r H n r, y) r!n r)! 30) which represents a kind of epansion of the ordinary monomials in terms of a Hermite-like family of polynomials, which can be proved to belong to an orthogonal set for negative values of y only. Let us consider a given function f) to be epanded in series of ordinary Hermite polynomials, namely f) = c n He n ) 31) according to eq. 8) we can also write ep 1 ) f) = The use of the Gauss transform, namely ) ep a f) = 1 ep πa allows the following conclusion. If the function Π) = 1 π ep c n n. 3) ) ξ) fξ)dξ 33) 4a ) ξ) fξ)dξ 34)
Monomiality, orthogonal and pseudo-orthogonal polynomials 609 can be epanded as Π) = c n n 35) then the epansion 31) in terms of Hermite polynomials hold, we find indeed from eq. 34) that n ) Π) = ep ξ He n ξ)fξ)dξ, 36) π which yields the general form of the c n coefficient and the orthogonality of the polynomials He n ). According to the previous results the orthogonality of Hermite polynomials is one of the consequences of their operational definition 8) and of the associated Gauss transform. Just to give a further eample we note that for f) = epb) we find Π) = epb +b) 37) which leads to c n = epb ) b)n 38) and thus to a well known epansion in terms of Hermite polynomials. In the case of the polynomials H n ) we can formulate our statement in analogous terms, by noting indeed that H n ) = ep 1 ) ) n 39) 4 we have to replace the function Π) with 1 ) Π) = ep f) = 1 4 π ep ξ) )fξ)dξ 40) and c n with c n = c n. 41) n In the case of f) = ep a ), a > 1, we find Π) = 1 ep a ). 4) 1+a 1+a Thus yielding c n = 1)n n a n 1 + a ) n+ 1 43)
610 G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci and the corresponding, well known, epansion in terms of Hermite polynomials. For further etensions of the above point of view to other polynomial sets see the concluding section. We must underline that, albeit we have not done any eplicit request of orthogonality, the present reformulation of the series epansion in terms of Hermite polynomials is perfectly equivalent to the ordinary case and the conditions underlying the epansions are left unchanged. It is worth stressing that not all the Hermite-Kampé de Fériet provide orthogonal sets, in the strict sense, they are indeed limited to negative values of the y variable only. Notwithstanding, according to eqs. 8)-33), we can obtain epansions in terms of Hermite-Kampé de Fériet polynomials even when orthogonality in the strict sense is not guaranteed. We find indeed ) 1 ep 1+4a 1+4a ) = ep a ep )= 1) n = H n, a), a < 1 44) 4. In the forthcoming section we will see how the above algebraic point of view can be etended to other families of polynomials. We can eploit the previous results to discuss the concept of pseudoorthogonality from a different point of view. We consider indeed the polynomials G n, y) defined as [ n] n r y r G n, y) = [n r)!] r! 45) which provide a family of polynomials discussed in ref. [1] and share some analogies with the Hermite polynomials in the sense that G n, y) =H n 1 ˆD,y). 46) It is therefore worth noting that they can be defined through the operational rule G n, y) = ep y ) n ), 47) which can be eploited in perfect analogy with the analogous rule defining the Hermite polynomials to decide on their orthogonality properties. To this aim we will try to establish the eistence of the analogous of the Gauss transform for the action of the operator ep [ α )] on a given
Monomiality, orthogonal and pseudo-orthogonal polynomials 611 function. To this aim we note that in operational terms, using the correspondence 1 ˆD 48) we get also see ref. [1]) = ˆD 1. 49) Let us now consider any function f) admitting the series epansion f) = and assume that, in correspondence of this, the function linked to f) by the Laplace-type transform g) = a r r! r 50) g) = a r r 51) + 0 ft) ep t)dt 5) eists. We can therefore state the following identity ep α ) f) = ep α ˆD 1 ) g ˆD 1 ). 53) Accordingly we can write that ep α ) f) = 1 ep πα 1 ˆD ξ) ) gξ)dξ. 54) 4α Let us now consider the epansion of the function f, y), where y>0 denotes a fied parameter, in terms of the polynomials G n, y), namely f, y) = c n G n, y). 55) By applying the same considerations, we used in the case of Hermite polynomials, we find that Π, y) := ep y ) f, y) = c n y) n 56)
61 G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci and, according to eq. 54), we and up with Π, y) = 1 1 ˆD ξ) ) ep gξ)dξ = πy 4y c n y) n, 57) thus getting c n y) = 1 ) ep ξ ξ H n πy y, 1 ) gξ)dξ. 58) 4y The conclusion is that the polynomials G n, y) are not orthogonal in the strict sense, but they can be eploited to epand a function f, y) using the coefficients of the epansion in terms of Hermite polynomials of the function g). The polynomials G n, y) can therefore be considered pseudoorthogonal and its partners, omitted for the sake of conciseness, have been discussed in ref. [5]. According to the discussion of this section, we can draw a general conclusion. We consider a family of polynomials p n ) defined by the operational rule p n ) = T n, 59) where the operator T admits an inverse. If the inverse function T 1 f) =Φ) 60) eists and can be epanded in Taylor series, then the function f) can be epanded in series of the above polynomials with the same coefficients of the Taylor epansion of the function Φ). 5 The Legendre polynomials It has been shown that the Legendre polynomials can be framed within the monomiality treatment and that they can be introduced using an operational definition in between the Hermite and the Laguerre case, we have indeed ref. [1]) ) [ S n, y) =C 0 y n n] 1) r y n r r = n r)!r!). 61) The ordinary Legendre polynomials are obtained as a particular case of 33) as for the ordinary formulation see refs. [1], []) ) 1 y P n y) =S n,y. 6) 4
Monomiality, orthogonal and pseudo-orthogonal polynomials 613 We can now eploit the same point of view as before and look for epansions in terms of Legendre polynomials with the only help of algebraic concepts. We can etend the previous considerations relevant to the Hermite polynomials to the present family by noting that an alternative operational definition is S n, y) =H n y, D 1 ). 63) In this case we can directly apply the formalism developed for Hermite polynomials in a fairly direct way. We can indeed epand the function fy) = epby) as follows epby) = b) n ep4b D 1 )H n y, D 1 ). 64) The above relation is of operational nature and can be written in a more eplicit form by eploiting the following identity H n y, D 1 ) ep4b D 1 ) = [ n ] 1) r y n r r C r 4b )= n r)!r! 65) = B n y, ;4b ), thus getting b) n epby) = B n y, ;4b ). 66) We will comment on the nature of the function B n, y) in the concluding section. By noting furthermore that ep4b D 1 )H n y, D 1 4b) r )= S r r! n, y), n [ Sn r ] 67), y) = 1) s y n s s n s)!r + s)!s! we can obtain an alternative epansion in terms of the polynomials S r n, y), whose link with the associated Legendre polynomials will be discussed in a forthcoming paper. Before concluding this section we will see how the ordinary monomials can be epanded in Legendre series. The use of the operational analogy with the Hermite polynomials and eq. 30) yields [ n ] y n = D r H ny, D 1) r!n r)! [ n ] = r S r n r, y) r!n r)! 68) again obtained without any assumption of orthogonality. In the forthcoming concluding section we will clarify some points left open in the so far developed analysis.
614 G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci 6 Concluding remarks In the previous section we have introduced the functions B n a, b; ) =Bn 0 a, b; ), without any further specification. We will now comment on their nature and discuss an independent derivation in terms of the generating function epat)c m bt ) = [ Bn ma, b; ) = t n Bm n a, b; ), n ] which can be easily derived after noting that epat)c m bt ) b r a n r n r)!r! C m+r) = ep at bt ) C m ) = t n = H n a, b ) C m ), ) s 1) s C m ) = C m+s). 69) 70) This family of functions satisfies the recurrences a Bm n a, b; ) = nbm n 1 a, b; ) b Bm n a, b; ) = nn 1)Bm+1 n a, b; ) 71) which are a direct consequence of their generating function. In the previous sections we have introduced the Laguerre polynomials by using the operational definition 7), according to which the variable is viewed as a parameter, on the other side we can use the identity see ref. [1]) ep y ) 1) n n ) = L n, y) 7) which is just a consequence of the fact that the Laguerre polynomials satisfy the p.d.e. F, y) = F, 0) = )n. F, y) 73)
Monomiality, orthogonal and pseudo-orthogonal polynomials 615 More in general if F, 0) = f) the solution of the above equation can be written in terms of the following integral transform ref. [5]) ) ) + F, y) = ep ep s)c 0 y 0 y s f ys)ds 74) which holds since the 0 th order Tricomi function is an eigenfunction of the operator. It is evident that the previous transform plays a role analogous to the Gauss transform of the Hermite case, we can therefore formulate the following statement: If the function + F ) = ep ) ep s)c 0 s)fs)ds 75) 0 admits the epansion ) n F ) = c n 76) then f) can be epanded in terms of Laguerre polynomials as f) = c n L n ). 77) By epanding, indeed, the eponential and the Tricomi function in eq. 75), we get ) r + s) p F ) = ep s) r! 0 p=0 p!) fs)ds = 78) ) n + = ep s)fs)l n s)ds, 0 which is the proof of our statement. In a forthcoming paper we will see how the present results can be etended to higher order and multidimensional Hermite polynomials. Before concluding we want to discuss a further problem involving the Chebyshev polynomials of second kind U n ), which are linked to the Hermite polynomials by means of the following transform see ref. [1]) + U n ) = 1 ep s)s n/ H n s)ds. 79) 0 If a given function f) can be epanded in terms of Chebyshev polynomials, we also have c + n f) = ep s)s n/ H n s)ds 80) 0
616 G. Dattoli, B. Germano, M.R. Martinelli and P.E. Ricci if the sum on the r.h.s. of eq. 65) is such that F, s) = we can conclude that all the functions of the type f) = + 0 c n sn/ H n s), 81) ep s)f, s)ds 8) can be epanded in terms of U n ) polynomials, just eploiting the corresponding epansion in terms of ordinary Hermite. The function F ) = ep ) corresponds to f) =1/1 ) and according to the previous relations and to eqs. 35)-38) we find 1 1 = 4 + 3 1 3 ) n ) U n 3. 83) This procedure is fairly helpful but does not fully reply the concept of orthogonality of Chebyshev polynomials, since the family of functions 69) is only a family of functions admitting an epansion in terms of U n ) polynomials. We will show elsewhere that the method can be etended to Legendre, Gegenbauer,... polynomials and that their orthogonality properties can be framed within an algebraic contet. References [1] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, Proc. of the Intern. Workshop on: Advanced Special Functions and Applications, Melfi 9 1 May, 1999, D. Cocolicchio, G. Dattoli and H. M. Srivastava eds.), Aracne Editrice, Roma, 000. [] L.C. Andrews, Special functions of Mathematics for Engineers, Oford Univ. Press, Oford New York, 1998. [3] P. Appell and J. Kampé de Fériet, Fonctions hypergéométriques et hypersphériques. Polynômes d Hermite, Gauthier-Villars, Paris, 196. [4] H.M. Srivastava and M. Manocha, A treatise on generating functions, Ellis Horwood, New York, 1984. [5] G. Dattoli, B. Germano and P.E. Ricci, Comments on monomiality, ordinary polynomials and associated bi-orthogonal functions, Appl. Math. Comput., 154 004), 19-7. Received: April 4, 005