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Operational Techniques for Laguerre and Legendre Polynomials

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Operational Techniques for Laguerre and Legendre Polynomials INTERNATIONAL JOURNAL of PURE MATHEMATICS Volume 2, 2015 Operational techniques for Laguerre and Legendre polynomials Clemente Cesarano, Dario Assante, A. R. El Dhaba Abstract— By starting from the concepts and the related n , (2) formalism of the Monomiality Principle, we exploit methods of δ xn =∏() xm − δ operational nature to describe different families of Laguerre m=0 polynomials, ordinary and generalized, and to introduce the Legendre polynomials through a special class of Laguerre polynomials whose associated multiplication and derivative operators read: themselves. Many of the identities presented, involving families of d different polynomials were derived by using the structure of the ^ δ operators who satisfy the rules of a Weyl group. M= xe dx In this paper, we first present the Laguerre and Legendre d δ −1 (3) polynomials, and their generalizations, from an operational point of ^ e dx P = . view, we discuss some operational identities and further we derive δ some interesting relations involving an exotic class of orthogonal polynomials in the description of Legendre polynomials. It is worth noting that, when δ = 0 then: Keywords— Laguerre polynomials, Legendre polynomials, Weyl group, Monomiality Principle, Generating Functions. ^ Mx= (4) ^ d I. INTRODUCTION P = . In this section we will present the concepts and the related aspects of dx the Monomiality Principle [1,2] to explore different approaches for More in general, a given polynomial Laguerre and Legendre polynomials [3]. The associated operational calculus introduced by the Monomiality Principle allows us to reformulate the theory of the generalized Laguerre and Legendre pxn (), nx∈∈, polynomials from a unified point of view. In fact, these are indeed shown to be particular cases of more general polynomials and can be can be considered a quasi-monomial if two operators also used to derive classes of isospectral problems, an interesting example is given by the generalized Bessel functions [4]. Many ^ ^ properties of conventional and generalized orthogonal polynomials M and P have been shown to be derivable, in a straightforward way, within an operational framework, which is a consequence of the Monomiality called multiplicative and derivatives operator respectively, can Principle. Before to investigate the case of the generalized Laguerre be defined in such a way that: and Legendre polynomials, let us briefly discuss about the Monomiality Principle. ^ By quasi-monomial [1] we mean any expression characterized by an Mp() x= p () x integer , satisfying the relations: nn+1 (5) n ^ P pnn() x= np−1 () x ^ Mfnn= f+1 (1) ^ with: P fnn= nf −1 ^ ^ ^^ ^^ ^ M,1 P=−= M P PM , (6) ^ ^ where M and P play the role of multiplicative and derivative operators. An example of quasi-monomial is provided by: ^ ^ ^ that is M , P and 1satisfy a Weyl group structure [5] with respect commutation operation. The rules we have just C. Cesarano and Dario Assante are with the Faculty of Engineering, established can be exploited to completely characterize the International Telematic University UNINETTUNO, Rome, Italy (phone: +39 ^ ^ 0669207675; e-mail: [email protected]). family of polynomials; we note indeed that, if M and P have A. R. El Dhaba is with the Department of Mathematics, Faculty of Science, Damanhour University, Egypt ISSN: 2313-0571 30 INTERNATIONAL JOURNAL of PURE MATHEMATICS Volume 2, 2015 a differential realization, the polynomial pxn ()satisfy the differential equation: ^ −n xn Daax = , where a≡ const. (14) n! ^^ MPpx()= px (). (7) nn and: If px()= 1, then px() can be explicit constructed as: −1 0 n ^^d Dx = 1 . dx ^ n M(1)= pxn ( ) . (8) By following an analogous procedure, it is possible to derive the explicit forms of the Laguerre polynomials, directly by If px()= 1, then the generating function of px() 0 n using the concepts and the formalism of the Monomiality can always be cast in the form: Principle. In fact, as for the generalized Hermite polynomials, since Lx0 ()= 1, we have [1,7,13]: ^ +∞ t n tM e(1)= ∑ pxn ( ) (9) n=0 n! n ^^−−1 n n s s , (15) Lxn ( )=−=− 1 Dxx (1)∑ ( 1)D (1) s=0 s where tt∈,| | < +∞ . In some previous papers [6-9], we have shown that different that is: families of Hermite polynomials are an example of quasi- monomial. In the case of two-variable Kampé de Fériet n (− 1) ssx Lx()= n ! . (16) polynomials [10,11]: n ∑ 2 s=0 (ns− )!( s !) n 2 yxrn−2 r It is possible to generalize the Laguerre polynomials by using Hn (, xy )= n !∑ , (10) the formalism of the Monomiality Principle; in fact, by starting =0 rn!(− 2 r )! r from the operational relations [1,13], we can set: we note that the associated multiplication and derivative −1 n operators, are identified as: ^ Ln (,) xy= y − Dx (1) (17) ^ ∂ Mx= + 2 y ∂ to define the generalized two-variable Laguerre polynomials. x (11) ^ ∂ By multiplying both sides of above expression with t n , P = . ∂x tt∈,| | < +∞ , and then summing up over n, we can state the link between the two-variable Laguerre polynomials and their According to what has been discussed above, since generating function: H0 (, xy )= 1 then Hn (, xy ) can be explicitly constructed as: n +∞ ^ −1 +∞ n nrt y−= Dx tL(, xy ) ∂∂ n n ∑∑ n r nn=00= x+=2 y (1)∑ (2y ) Hnr− ( xy , ) (1) = ∂∂xx r =0 r (12) = Hn ( xy , ). and, finally: The above identity is essentially the Burchnall operational 1 xt +∞ exp −=n (,) (18) formula, whose proof can be found in the paper [12]. ∑tLn xy 11−−yt yt n=0 The Laguerre polynomials too, can be viewed as quasi- monomial [1,13] and the relevant characteristic operators are: which immediately gives the following relation: ^^−1 MD=1 − x n x (13) Lnn(, xy )= yL. (19) ^ ∂∂ y Px= − ∂∂xx It is important to emphasize the powerful tool represented by ^ −1 the formalism of the Monomiality Principle. Moreover, we can with Dx being a negative derivative operator defined as: ISSN: 2313-0571 31 INTERNATIONAL JOURNAL of PURE MATHEMATICS Volume 2, 2015 derive other important relations, linking the ordinary Laguerre II. OPERATIONAL IDENTITIES FOR LAGUERRE POLYNOMIALS polynomials and their generalizations of two variables. AND LEGENDRE POLYNOMIALS It is easy to obtain the following recurrence relations regarding In the previous section, we have introduced the ordinary the generalized two-variable Laguerre polynomials [1,3]: Laguerre polynomials Lx() by using the formalism of the n Monomiality Principle (see eq. (15)). It is immediately to note ∂ Lnn(, xy )= nL−1 (, xy ), (20) that: ∂y nn ^^−−11n n ^−s s (25) ∂∂ L2nn( x )=−−=− 1 Dxx 1 D (1)∑ ( 1)Dx Lx ( ) x+= y Lnn(, xy ) nLxy (, ), (21) s=0 s ∂∂xy and by applying the identities outlined in equations (15) and (16) and by observing the explicit form of the associate and finally: Laguerre polynomials, presented in equation (24), we easily obtain that: 2 ∂∂ ∂ −+x2 y Lnn(, xy ) = L (, xy ). (22) ∂ ∂∂ ^^−−ssn kk s x yy (− 1) x nx! ()s Dxx Lxnn()= D n ! 2 = L() x (26) ∑ k =0 (nk− )!( k !) (ns+ )! The last relation could be seen as a first order linear differential equation in the variable y of the polynomials which allows us to conclude with: function L(, xy ). By noting that: n n ss 2 (− 1) x ()s +∞ +∞ n Lx2nn()= (!) n Lx(). (27) n nn(− 1) ∑ s=0 (n−+ s )!( n ss )! ! ∑∑t Ln ( x ,0)= exp() −= xt t x , nn=00= n! It is also possible to write the ordinary Laguerre polynomials we get: of index 2n in a different form. We begin to note that: n (− 1) −−12n n ^^ Lxn ( ,0) = x (23) n! Lx2n ( )=−+ 1 2 Dxx D (1) , and so that, we can write then solution of the Cauchy’s problem of equation (22) with initial condition y = 0 : which, after exploited the above formal identity, we get: ns− 2 n ^^−−21s ∂∂ n −+yx y n 2 () = xx1− 2 (1) . ∂x ∂y (− 1) n Lx2n ∑ D D Ln (, xy )= e x. s=0 s n! It is also possible to define the so-called associate generalized It is easy to recognize that, starting from the operational Laguerre polynomials through the use of the Monomiality definition of the ordinary Laguerre polynomials (see equation Principle. By following the procedure followed to introduce (16)), we can obtain: the ordinary Laguerre polynomials (see eq. (15)), we have n [3,13]: ^ −1 1−=µµ (1) ( ) , (28) Dx Lxn ααn + ^ −1 n nn ()α d dx(− 1) Lxn ()1=−−=− 1Dx (1)1 (24) dx dx n! where µ is a continuous variable. The expression of the Laguerre polynomials Lx2n (), state in relation (27), can be with α being not necessarily an integer. now written in the following form: The use of the above operational identities could be used to simplify the derivation of many properties concerning the n 2s x (2s ) Laguerre and generalized Laguerre polynomials. In the next Lx2n()!= n∑ Lns− (2 x ) . (29) =0 (n+ ss )! ! section we will derive different relations involving these s families of polynomials and we also start to introduce the Legendre polynomials through a special class of generalized The procedure shown above could be generalized to derive Laguerre polynomials. further interesting operational relations involving the ordinary Laguerre polynomials, in particular to obtain the addition and ISSN: 2313-0571 32 INTERNATIONAL JOURNAL of PURE MATHEMATICS Volume 2, 2015 multiplication formulae. We have, in fact, by using newly the ^^ operational definition of the polynomials (), given in MP,1= Lxn equation (16): we can immediately obtain the explicit form of the nm ^^−−11n n ^−s polynomials defined by the above operators s , Lnm+ ( x )=−−=− 1 Dxx 1 D (1)∑ ( 1)DLxxm ( ) =0 s s n n ^^n −1 ∂ 2 xyrnr−2 (1)=+= 2 (1) ! (34) and then: M yDx n∑ 2 ∂y r =0 (n− 2 rr )!( !) n (− 1) ss x ()s where again we have used the identity: Lnm+ () x= nm ! !∑ Lm() x , (30) s=0 (n−− s )!( m ss )! ! ^ −1 xn that is the summation formula.
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