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 ^ Mf= f nn+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
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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]: ^ +∞ n tM t 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) = ∂∂∑ (12) xx r =0 r
= 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 Lx()= D n ! = L() x (26) nn∑ 2 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 (, ). By noting that: Ln xy 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. Similarly, we can easily state Dx = . (35) n! the multiplication formula, by following the same procedure.
We have, in fact: The polynomials obtained in relation (34) can be recognized as
a special class of Laguerre polynomials [13-15] and then we n n (−− 1)ss[] (m 1) nx ! ()s . (31) can set: Lxnm ()= ∑ Ln( m− 1) () x s=0 s []nm(−+ 1) s ! n 2 xyrnr−2 The operational treatment shown in the previous sections, (, )= ! . (36) 2 Ln xy n∑ 2 regarding the Hermite and Laguerre polynomials, ordinaries r =0 (n− 2 rr )!( !) and in some cases generalized, can be also used to investigate, under an operational point of view, different families of A fortiori, we note that the condition prescribed by the classical polynomials, as for example the Legendre and Jacobi Monomiality Principle is respected, in fact: polynomials. Moreover, this approach allows us to derive new identities for these classes of polynomials and, by using the 20L(, xy )= 1. (37) formalism of the Monomiality Principle, helps us to recognize many other families of polynomial as quasi-monomial. These generalized Laguerre polynomials can be reduced to the We remind that the generalized Hermite polynomials ordinary Legendre polynomials:
Hn (, xy ) and the two-variable Laguerre polynomials Ln (, xy ) n can be viewed as quasi-monomials with the following 2 (−− 1)nrr−−2xx (1 22 ) nr operators: ()= ! (38) Pxn n∑ 2 2(nr− 2 ) r =0 (n− 2 rr )!( !) 2 ^ ∂ ^^−1 MH = xy + 2 ∂ MLx= yD − x , (32) by setting: ^ ∂ ^ ∂∂ PH = PxL = − ∂x ∂∂xx 1 2 2 Lnn−−(1 x ), x = Px ( ) . (39) 4 It has been shown that the Legendre polynomials [14] can be introduced directly by using the multiplication and derivative We can also derive the generating function of the above operators acting on the Hermite and Laguerre polynomials, in polynomials. Since we have observed that 20L(, xy )= 1, we the sense that we can combine the above operators (eq. (32)) immediately get: to define new operators that can be used to view the Legendre polynomials as quasi-monomials and at same time to define ^ +∞ n them. Let the operators: t exptM (1)= ∑ 2 Ln ( x , y ) , (40) n=0 n! ^^−1 ∂ My= + 2 Dx ∂y where again tt∈,| | < +∞ . By exploiting the exponential on (33) ^ ∂ the above relation, we have: P = ∂y ^^−−11∂∂ ^ expt M (1)=+=+ exp t y 2 Dxx (1) expty 2 t D (1) by using the formalism of the Monomiality Principle [1,7-9], ∂∂yy after noting that: Since the operators do not commute:
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^^−−11 and by following the same procedure adopted in the case of ∂ 2 ty,2 t Dxx= −2 t D , polynomials Lx(), we get: ∂y n
n and then we obtain: s n Rn( xy , )= n !∑ ( − 1)Lns− ( xL ) s ( y ) . (46) s=0 s ^−1∂∂ ^^−−11 2 expt y+= 2 Dx (1) exp()ty exp 2 Dxx exp t D (1) These polynomials could be also recognized as quasi- ∂∂yy monomial if we consider the following form:
R(, xy ) n n that is: n s =∑( − 1)Lns− ( xL ) s ( y ) , (47) n! s=0 s
^−1 ^^−−11 ∂∂ 2 expt y+= 2 Dx (1) exp()ty exp t Dxx exp 2 D (1) and the operators: ∂∂ yy ^ ^^−−11 M=−+ DDxy From the fact that: ^ ∂∂ Pxx = − (48) ∂∂xx −1 ^ ∂ ^ ∂∂ exp 2Dx (1)= 1 , Pyy = ∂y ∂∂yy we finally obtain: where is immediately verify the rules of Monomiality Principle, presented in previous section. By following the
^ −1 +∞ n 2 above procedure, we can also derive the related generating yt Dx t t ee = 2 L(, xy ). (41) function: ∑ ! n n=0 n n +∞ ttnn+∞ ^^−−11 We can observe that the Tricomi function of order zero [16] of (, )= −+ = ∑∑2 Rn xy Dxy D a suitable variable, reads: nn=00(n !) = n! (49)
^^−−11 tDyx− tD +∞ 2nn tx =e e = C00( − yt ) C ( xt ). −=2 (42) C0 () xt ∑ n=0 nn!! so, we can write: We note that the generating function of the special polynomials defined in equation (45) could be expressed in +∞ n terms of Tricomi functions of zero order as for the generalized yt 2 t Laguerre polynomials of type (, ) (see equation (43)). e C02()−= xt Ln (, x y ) (43) 2 Ln xy ∑ ! n=0 n The procedure followed to introduce the polynomials: and then, relatively to the Legendre polynomials, by using Rn (, xy ) equation (39), we can state: Rn (, xy ) or (50) n!
+∞ t n 1 xt 22. (44) and the similarity of their generating function with the ∑ Pn ( x )= eC0 (1 − x ) t n=0 n!4 polynomials 2 L(, xy ), lead us to observe that it is possible to n derive the Legendre polynomials in a different way. From The use of the structure of Laguerre polynomials, recognized equation (18), we can write: as , allowed us to define the Legendre quasi-monomials polynomials, in this case through a special class of Laguerre. nr− r − 11−+xx We can follow a different approach, by considering the (− 1) nr n ordinary Laguerre polynomials introduced in the previous 2 22 Px( )= ( n !) (51) section. From equation (16), let us introduce this family of n ∑ 2 2 r =0 [](nr− )! ( r !) polynomials [14,15]:
n and then, we find: ^^−−11 n ns−− s ns 2 (− 1) yx R(,)! xy=−+ n Dxy D =( n !) , (45) n ∑ 2 2 s=0 [](ns− )! ( s !)
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11−+xx polynomials Rn (, xy ). In fact, by using their operational Pxnn()= R , . (52) 22 definition, exploited in equations (45) and (46), we can, for example, derive a summation formula. We note that: These two sections have been shown how the operational n methods, inducted by the concepts of the Monomiality (2n )!^^−−11 Principle, provide to determine a synthetic and R2nn(, xy )= −+ Dxy D R(, xy ), (55) n! computationally powerful view on the theory of one variable and multi-variable polynomials, in particular in the treatment of some generalizations of the polynomials belonging to the by exploiting the r.h.s of the above identity, we get: Laguerre family and in the description of the special nature of the Legendre polynomials. In the next section, we will derive (2n )! n n ^^−−()ns − s (, )= (1) − s (, ), (56) some useful identities regarding the Legendre polynomials, by R2nn xy ∑ Dyx D R xy n! s=0 s proceeding as done before for the Laguerre polynomials.
and finally, after noted the expression of the associate polynomials of the form (,)pq (, ), presented in equation III. OPERATIONAL RULES FOR LEGENDRE POLYNOMIALS Rn xy (53), we have: The method based on the rules of Monomiality Principle, described in the previous sections, is devoted to simplify the n ns− s operational relations of many families of classical polynomials sn yx (,sn− s ) R2nn(, xy )= (2)!! n n∑ (1) − R(, xy ). and, from the other hand, to recognized generalizations of s=0 s (2ns−+ )!( ns )! well-known classes of polynomials or special functions, by using the identities derived in the descriptions of those In Section II, we have defined the generalized two-variable polynomials themselves. This method could be used in the Laguerre polynomials of the type 2 L(, xy ) to derive the future to explore many interesting topics such as those n Legendre polynomials. It is immediately to exploit a different explored for damage mechanics [17], for the dynamics of generating function: contact [18] and for electromagnetic field problems [19-20], were series representations are important solution methods. In +∞ 1 this section, we will extend the results obtained for the n t2 Ln (, xy )= (57) ∑ 22 Hermite and Laguerre polynomials to the family of Legendre n=0 12−+−yt() y4 x t polynomials by using the characterization introduced in
Section II and will also discuss some useful relations, linking where is always ∈,| | < +∞ . By noting that the Legendre the above mentioned classes of orthogonal polynomials. tt We have introduced in Section I, the associate ordinary polynomials are linked to the following generating function: Laguerre polynomials, by acting directly on the operational +∞ realization. It is possible to extend such a generalization n 1 tP() x= , (58) defined in equation (45), ∑ n 2 regarding the polynomials Rn (, xy ) n=0 12−+xt t directly on their explicit forms.: we immediately can conclude that: n (− 1) syx ns− s R(,)pq ( xy , )=++ ( n p )!( n q )! . n ∑ n s=0 (ns− )!( nsqssp −+ )! !( + )! y 2 2 2 Lnn(, xy )=() y − 4 x P . (59) (53) yx2 − 4 Let us remind that the inverse derivative operator acts in the following way: The above relations between the generalized Laguerre polynomials and Legendre polynomials, allows us to derive −n other operational relations involving the polynomials R(, xy ) ^ xn n Daax = , where a≡ const. and links with the Legendre polynomials too. By comparing ! n the previous equations (39) and (59),we have in fact:
and then, we can state the following identity: n yx− Rnn(, xy )=() x + y P, (60) ^^−−pq 2 pq yx+ (n !) xy (,)pq Dxy D R(, xy )= R(, xy ). (54) nn(np++ )!( nq )! which stresses the link shown in equation (52). From equation
(57), we can use the above identity to deriving the generating It is possible to obtain similar relations outlined for the function of the polynomials R(, xy ). We have in fact: ordinary Laguerre polynomials, as done in Section I, for the n
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+∞ n n 1 2 n−2 rr tRn (, xy )= . (61) ∑ 22 ()s yx =0 ( , )= ( + )! . n 1−−++ 2(x yt ) () x y t 2 Ln xy n s ∑ r =0 (n−+ 2 rrr )! !( s )!
The relations presented in Section II, regarding the generating The above relations are useful to state the following important function of the Laguerre polynomials of type 2 Ln (, xy ) (eq. identity: (43)), can be used to obtain a further generating function for the polynomials R(, xy ), that is: n s ^ −1 n ()s 2 ss() L(, xy )= n ! H yD,x x L (, xy ), (64) 22n ∑ ns−−ns s=0 (n− ss )! ! +∞ t n ()y− xt 2, (61) ∑ Rn (, x y )= e C0 () xyt n=0 n! which can be written in a more convenient form: which confirms the strong link with the special class of ^ −1 ,()s (, )= Laguerre polynomials L(, xy ) and then with the Legendre Hmn yDx 2 L xy 2 n polynomials. m (65) 2 m−2 ss yx ()s = mn!!∑ 2 Ln ( xy , ). =0 (m−+ 2 s )!( n ss )! ! IV. CONCLUDING REMARKS s
The results shown in the previous sections have described These last identities as well as those previously discussed in the Legendre polynomials through different classes of Section II, involving polynomials of two indices, suggest that polynomials. The Laguerre-like polynomials of the type these families of polynomials can be viewed formally as two- (, ) and the exotic polynomials (, ). It is 2 Ln xy Rn xy index polynomials and then we expect that their generating interesting to note that many of the properties obtained were functions will be defined on a couple of indices instead of a derived by the use of the concepts and the related formalism of single index. the Monomiality Principle, in the sense that we have stated the identities related to the classes of polynomials discussed, by REFERENCES exploiting their nature of quasi-monomials. In this concluding [1] Cesarano, C., Monomiality Principle and related operational techniques section, we will present some relations which could be deeper for Orthogonal Polynomials and Special Functions, Int. J. of Pure investigated in forthcoming papers and that could be used in Mathematics, 1, pp. 1-7 (2014). modeling advanced materials in continuum mechanics [21-24], [2] Smirnov, Y. and Turbiner, A., Hidden SL2-Algebra of finite difference equations, , A10 pp 1795-1801 (1995). and in electromagnetic field propagation problems [25-27] and Mod. Phys. Lett [3] Andrews, G.E., Askey, R. and Roy, R., Special Functions, Cambridge in transmission lines [28-30]. University Press, (1999). We note that, by using the explicit form of the generalized [4] Cesarano, C., Assante, D., A note on generalized Bessel functions, two-variable Hermite polynomials (see equation (10)) and by International Journal of Mathematical Models and Methods in Applied , 7 (6), pp. 625-629, (2013). the definition of the Laguerre-like polynomials (, ), we Sciences 2 Ln xy [5] Miller, W., Lie Theory and Special functions, Academic Press, New can write [14]: York, (1968). [6] Dattoli, G., Cesarano, C., On a new family of Hermite polynomials associated to parabolic cylinder functions, Applied Mathematics and −1 ^ , 141 (1), pp. 143-149 (2003). (, )= , , (62) Computation 2 Lnn xy H yDx [7] Cesarano, C., A note on Generalized Hermite polynomials, Int. J. of Applied Mathematics and Informatics, 8, pp. 1-6, 2014 [8] Cesarano C., Hermite polynomials and some generalizations on the Heat by implying that the action of the operator contained in the equations, Int. J. J. of System Applications, Engineering & Development, 8, pp. 193-197, (2014) argument of the Hermite polynomials is referred to the unit. [9] Cesarano, C., Cennamo, G.M. and Placidi, L., Humbert Polynomials The use of the inverse derivative operator allows us to state and Functions in Terms of Hermite Polynomials Towards Applications useful operational identities related to the Laguerre-like to Wave Propagation, WSEAS Transactions on Mathematics, 13, pp. polynomials. We have, in fact: 595-602, (2014). [10] Appell, P., Kampé de Fériet, J., Fonctions hypergéométriques et hypersphériques. Polinomes d’Hermite, Gauthier-Villars, Paris, (1926). ^ −s s nx! ()s [11] Gould, H.W., Hopper, A.T., Operational formulas connected with two Dx L(, xy )= L(, xy ), (63) 22nn( )! generalizations of Hermite polynomials, Duke Math. J., 29 pp. 51-62 ns+ (1962). [12] Burchnall, J.L., A note on the polynomials of Hermite, Quarterly ournal where we have considered the associate to the Laguerre-like of Mathematics, os-12 (1), pp. 9-11, (1941). polynomials, as in equation (24), by setting: [13] Dattoli, G., Srivastava, H.M. and Cesarano, C., On a new family of Laguerre polynomials, Atti dell’Accademia delle Scienze di Torino, Classe di Sci. Fis. Mat. Nat., 132, pp. 223-230, (2000). [14] Dattoli, G., Srivastava, H.M. and Cesarano, C., The Laguerre and Legendre polynomials from an operational point of view, Applied Mathematics and Computation, 124, pp. 117-127, (2001).
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[15] Dattoli, G., Ricci, P.E. and Cesarano, C., A note on Legendre Clemente Cesarano has published two books and over than fifty papers on polynomials, AInt. J. of Nonlinear Science and Numerical Simulation, international journals in the field of Special Functions, Orthogonal 2, pp. 365-370, (2001). Polynomials and Differential Equations. [16] Srivastava, H.M., Manocha, H.L., A treatise on Generating Functions, Wiley, New York, (1984) Dario Assante received the Laurea degree (Hons.) and the Ph.D. degree in [17] Placidi, L., A variational approach for a nonlinear 1-dimensional second electrical engineering from the University of Naples Federico II, Naples, Italy, gradient continuum damage model, Continuum Mechanics and in 2002 and 2005, respectively. Thermodynamics, p. 1-16, (2014). He is currently Assistant Professor of Electrical Engineering at the [18] Andreaus, U., Chiaia, B., Placidi, L., Soft-impact dynamics of International Telematic University Uninettuno. He is editor of technical deformable bodies, Continuum Mechanics and Thermodynamics, vol. journals in the field of electrical engineering. His research interests are related 25, p. 375-398, (2013). to the electromagnetic modeling, EMC, lightning effects on power lines, [19] Assante, D., Davino, D., Falco, S., Schettino, F., Verolino, L., Coupling modeling of high-speed interconnects, particle accelerators, and engineering impedance of a charge traveling in a drift tube, IEEE Transactions on education. He is coauthor of more than 30 papers published in international Magnetics, 41 (5), pp. 1924-1927, (2005). journals and conference proceedings. [20] Assante, D., Verolino, L., Efficient evaluation of the longitudinal coupling impedance of a plane strip, Progress In Electromagnetics A. R. El Dhaba is assistant Professor of Applied Mathematics at Damanhour Research M, 26, pp. 251-265, (2012). University since 2010. He was awarded the Ph. D. degree in Applied [21] Neff, P., Ionel-Dumitrel, G., Madeo, A., Placidi, L., Rosi, G., A unifying Mathematics from the Faculty of Science, Cairo University in 2009 in the perspective: the relaxed linear micromorphic continuum, Continuum field of Thermo-magneto-elasticity. His main research interests are: Mechanics and Thermodynamics, vol. 26, p. 639-681, (2014). Continuum Mechanics of Electromagnetic Media and Boundary Integral [22] Madeo, A., Rosi, G., Placidi, L., Towards the design of meta-materials Methods in the Theory of Thermo-elasticity and Thermo-magneto-elasticity. with enhanced damage sensitivity: second gradient porous materials, Starting from September 2014, the current position is a Post-Doctorate at the Research In Nondestructive Evaluation, vol. 25, p. 99-124, (2014). Faculty of Engineering – International Telematic University UNINETTUNO [23] dell’Isola, F., Madeo, A., Placidi, L., Linear plane wave propagation and – Roma. normal transmission and reflection at discontinuity surfaces in second gradient 3D Continua, ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 92 (1), pp. 52-71, (2012). [24] Abo-el-nour, N. A., Aishah, R., Placidi, L., The influence of initial hydrostatic stress on the frequency equation of flexural waves in a magnetoelastic transversly isotropic circular cylinder, ZAMM – Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, submitted. [25] Andreotti, A., Assante, D., Mottola, F., Verolino, L., Fast and accurate evaluation of the underground lightning electromagnetic field, IEEE International Symposium on Electromagnetic Compatibility, (2008). [26] Andreotti, A., Assante, D., Pierno, A., A. Rakov, V., Rizzo, R., A comparison between analytical solutions for lightning-induced voltage calculation, Elektronika ir Elektrotechnika, 20 (5), pp. 21-26, (2014). [27] Assante, D., Cesarano, C., Simple semi-analytical expression of the lightning base current in the frequency-domain, Journal of Engineering Science and Technology Review, 7 (2), pp. 1-6, (2014). [28] Assante, D., Andreotti, A., Verolino, L., Considerations on the characteristic impedance of periodically grounded multiconductor transmission lines, IEEE International Symposium on Electromagnetic Compatibility, (2012). [29] Andreotti, A., Assante, D., Rizzo, R., Pierno, A., Characteristic impedance of periodically grounded power lines, Leonardo Electronic Journal of Practices and Technologies, 12 (22), pp. 71-82, (2013). [30] Andreotti, A., Assante, D., Verolino, L., Characteristic impedance of periodically grounded lossless multiconductor transmission lines and time-domain equivalent representation, IEEE Transactions on Electromagnetic Compatibility, 56 (1), pp. 221-230, (2014).
Clemente Cesarano is assistant professor of Mathematical Analysis at Faculty of Engineering- International Telematic University UNINETTUNO- Rome, ITALY. He is coordinator of didactic planning of the Faculty and he also is coordinator of research activities of the University. Clemente Cesarano is Honorary Research Associates of the Australian Institute of High Energetic Materials. His research activity focuses on the area of Special Functions, Numerical Analysis and Differential Equations. He has work in many international institutions as ENEA (Italy), Ulm University (Germany), Complutense University (Spain) and University of Rome La Sapienza (Italy). He has been visiting researcher at Research Institute for Symbolic Computation (RISC), Johannes Kepler University of Linz (Austrian). He is Editorial Board member of the Research Bulletin of the Australian Institute of High Energetic Materials, the Global Journal of Pure and Applied Mathematics (GJPAM), the Global Journal of Applied Mathematics and Mathematical Sciences (GJ-AMMS), the Pacific-Asian Journal of Mathematics, the International Journal of Mathematical Sciences (IJMS), the Advances in Theoretical and Applied Mathematics (ATAM), the International Journal of Mathematics and Computing Applications (IJMCA).
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