International Mathematical Forum, 5, 2010, no. 14, 649-662 Appèl Polynomial Series Epansions G. Dattoli ENEA UTS Fisiche Avanzate Centro Ricerche Frascati C.P. 67-00044 Frascati, Rome, Italy Dattoli@frascati.enea.it K. Zhukovsky Dept. of Optics and Spectroscopy, Faculty of Physics M. V. Lomonosov Moscow State University Moscow 119991, Russia zhukovskii@enea.it Abstract We reconsider the theory of Appèl polynomials using an operational point of view. We introduce the Appèl complementary forms and show that they can be eploited to derive general criteria to get the epansion of a given function in terms of the above polynomial families. We discuss the case of multi-inde Appèl polynomials and touch on the Sheffer families. Keywords: Appél polynomials, Sheffer polynomials, series epansion, orthogonal polynomials, multi-inde polynomials 1. Introduction The Appèl polynomials a n () are defined through the following generating function [1]: t n n! a n() =A(t)e t, (1) where A(t) is left, at the moment, unspecified, but it is assumed that there eists a finite region of t, in which the epansion of A(t)in Taylor series converges.
650 G. Dattoli and K. Zhukovsky The use of the obvious identity te t = ˆD e t, ˆD = d (2) d and the previous assumption that A(t) can be epanded in Taylor series, allows to recast eq. (1) in the following operational form: t n n! a n() =A( ˆD )e t. (3) Therefore, epanding the eponential in eq. (3) in series and by equating the each term of the t- power in the r.h.s. and the l.h.s. we end up with the following definition of the polynomials a n (): a n () =A ( ˆD ) n. (4) In what follows we will refer to A ( ) ˆD as to the Appèl operator and we will assume that, along with it the inverse of the Appèl operator [ A ( )] 1(see ˆD [2]) can be defined in such a way that [ A( ˆD ) ] 1 A( ˆD )=ˆ1. (5) Note, for eample, that in the case of the two-variable Hermite polynomials the identity (4) specializes as follows [3]: H n (, y) =e y ˆD 2 n, H n (, y) =n! [n/2] n 2r y r (n 2r)!r!, (6) while for the Truncated Eponential Polynomials (TEP) we get [3] ē n () = 1 1 ˆD n, n ē n () =n! The operational definition (4) can now be eploited to derive the main properties of the Appèl polynomials. Indeed, by taking the derivative of both sides of the eq. (4) with respect to, we obtain, on account of the operators A ( ˆD ) and ˆD commute 1 : r r! (7) 1 We have denoted by operatorsa, B. [Â, ˆB] = Â ˆB [ A( ˆD ), ˆD ] = 0 (8) ˆBÂa commutation bracket between the two
Appèl polynomial series epansions 651 the well known property Furthermore, from the (4) we write ˆD a n () =na n 1 (). (9) a n+1 () =[A( ˆD )] n (10) and then, according to eq. (4) we can introduce the multiplicative operator ˆM (see [2]) in the following way: a n+1 = ˆMa n () (11) Thus, with ˆM specified in terms of the Appèl operator as given below: the usage of the identity ˆM = A ( ˆD ) A ( ˆD ) 1, (12) [ f( ˆD ), ] = f ( ˆD ), (13) where f is the derivative of the function f, yields: ˆM = + [ A( ˆD ) ] 1 A ( ˆD ). (14) The operators ˆM and ˆD realize the multiplicative and derivative operators for the Appèl polynomial family, which, therefore can be viewed as the family of quasi monomials. It is also evident that the set of operators ˆM, ˆD and ˆ1 realizes the Weyl-Heisenberg algebra. Furthermore, by noting that ˆM ˆD a n () =na n () (15) we find that the Appèl polynomials satisfy the following equation, involving the derivation operator ˆD, which we will call the differential equation for Appèl polynomials: ˆD a n ()+ A ( ˆD ) A( ˆD ) ˆD a n () =na n (), (16) where A is the derivative of A. This equation is valid for all of the polynomials, belonging to the Appèl family.
652 G. Dattoli and K. Zhukovsky In the particular case of the Hermite polynomials we obtain the multiplicative operator ˆMand the differential equation for the Appèl polynomials as follows: ˆM = +2y ˆD 2y ˆD 2H n(, y)+ ˆD H n (, y) =nh n (, y), (17) whereas for the Truncated Eponential Polynomials we obtain: ˆM = + 1 1 ˆD, ˆD 2 ēn() ( + n) ˆD ē n ()+nē n () =0. (18) Further eamples will be discussed in the concluding section. Albeit fairly elementary, the above illustrated formalism lets us reach important conclusions. Indeed, note that the Appèl polynomials satisfy the following recurrence a n+1 () =( + A ( ˆD ) A( ˆD ) )a n(). (19) Being orthogonal polynomials, they are also characterized by the identity o n+1 () =(a n + b n )o n ()+c n o n 1 () (20) and thus, taking into account eqs. (9) and (17), we can argue that among the classical Appèl polynomials the Hermite family is the only possible candidate for orthogonal set. The concept of orthogonal polynomials has been recently reconsidered and pseudo orthogonal forms have been introduced [4]. In the forthcoming section we will see how series epansions in terms of the a n () polynomials can be defined in general terms, without any eplicit use of their orthogonal nature. 1. The Complementary Appèl polynomials and series epansion Let us define the family of polynomials a n(): a n() = [ A( ˆD ) ] 1 n (21) through the inverse of the Appèl operator. Let us call this family the a n () complementary. They belong to the Appèl family and they satisfy the following recurrences: ˆD a n() =na n 1(), ( A ( ˆD ) A( ˆD ) ) a n() =a n+1(). (22)
Appèl polynomial series epansions 653 It is easily checked that the complementary Hermite polynomials are epressed in terms of the ordinary Hermite polynomials simply as H n (, y) while the complementary TEP are ē n() = n n n 1. Let us assume that a given functionf(), admitting a Taylor epansion around the origin, can be epanded in Appèl series too, namely: f() = α n a n (). (23) Our goal is now to determine the epansion coefficients α n. According to eq. (4) we can write the epansion (23) in the form 2 f() =A( ˆD ) α n n, (24) which can be inverted according to the following rule: [ A( ˆD ) ] 1 f() = α n n (25) and from the common Taylor epansion of the function f() = can conclude by eploiting the eq. (21) that f (r) (0) a r! r() = f (n) (0) n! n we α n n. (26) The use of the orthogonality of the circular functions allows the derivation of the coefficients α n from the identity α n = β r,n = 1 2π f (r) (0) 2π 0 r! β r,n, e inϕ a r(e iφ )dφ. (27) In the case of the Hermite polynomials we find: 0; r n odd β r,n = r n ( y) 2 r! n!( ; r n even, r n 2 )! (28) 2 The identity α n A( ˆD ) n = A( ˆD ) α n n is justified by the assumption that the Appèl operator can be epanded in Taylor series and by the consequent assumption that the series α n n and m n! α n (n m)! n m converge uniformly.
654 G. Dattoli and K. Zhukovsky A cylindrical Bessel function can be epanded in the following series of the Hermite polynomials with the coefficients α n : J m () = ( ) m 2 α n = r=n ( 1) r (m+r)! β r,n r! α n H n ( 2 4,y),, (29) where β r,n are given by (28). For the epansion of a cylindrical Bessel function in terms of TEP polynomials we find for the coefficients β r,n β n,r = δ n,r rδ n,r 1 (30) and therefore the epansion of the J n () in TEPs reads as follows: J n () = ( ) n 2 s=0 e n () = n r r! ( ) ( 1) s (n+s+2) e 2 (n+s+1)! s 4. (31) Now let us consider the Euler polynomials E n (), which are defined by the generating function [1], [4]: t n n! e n() = 2et, t <π. (32) e t +1 Then the relevant Appèl operator and its inverse have the following form: A( ˆD )= 2 e ˆD+1, A( ˆD ) 1 = e ˆD+1 2. (33) We also obtain substituting the above epression for the Appèl operator into (21) the Complementary Euler polynomials e n () =1 2 [( +1)n + n ]. (34) Thus, we write the epansion of a given function in series of the Euler polynomials: f() = α n e n (), α n = f (r) (0) r. (35) +1 r=n r! n The TEP are often eploited in optics to treat the propagation of the so called flattened beams [5]; recently the present author has eploited higher order
Appèl polynomial series epansions 655 TEP to study the propagation of beams having the particular configuration a transverse distribution with a hole [6]. The higher orders TEP: ē n ( m) = 1 1 ˆD m [n/m] n n mr = n! (n mr)! (36) are Appèl polynomials too. The associated complementary forms are ē n(n m) = n n! (n m)! n m. (37) For reasons we will clarify in what follows, we are mainly interested in epressing the ē n ( m) in terms of the higher order Hermite polynomials, which can be defined by the following operational identity: H n (m) (, y) = ep(y ˆD [n/m] m ) n n mr y r = n! (n mr)!r!. (38) With the help of the Laplace transform identity we write : 1 1 ˆD m = 0 e s e s ˆD m ds (39) and then we derive accounting for (36) and (38) the following integral form for the TEP: ē n ( m) = 0 e s H (m) n (, s)ds. (40) The above formula (40), along with the definition of the Hermite polynomials (38) suggest that e τ ˆD m H n (m) (, y) =H n (m) (, y + τ). (41) Thus the propagation property of the higher order TEP follows: e τ ˆD m ēn ( m) = 0 e s H (m) n (, s + τ)ds. (42) We will comment on the usefulness of the above epansions in the concluding section. 3 Appèl and Laguerre Polynomials
656 G. Dattoli and K. Zhukovsky The introduction of an etra variable in the theory of classical polynomials is particularly useful for a number of reasons and even though they can be reduced to the standard form by means of a straightforward transformation 3, an additional variable is the key tool to get e. g. new operational definitions. Regarding this point we must take some cautions when we are framing classical polynomials within the contet of the Appèl family. The two-variable Hermite polynomials can be considered as members of the Appèl family with respect to the y variable only. The two variable Laguerre polynomials n ( 1) r y n r r L n (, y) =n! (n r)! (r!) 2 (43) reduce to ordinary Laguerre polynomials for y = 1 are, indeed, not Appèl polynomials with respect to. The two variable Laguerre polynomials can indeed defined in terms of the following operational identity: L n (, y) =e y L ˆD [ ] ( ) n n!, ˆD L = ˆD ˆD. (44) It can be easily checked by noting the action of the Laguerre derivative L ˆD on ( )n : n! [ ] [ ] ˆD ( ) n ( ) n 1 L = n. (45) n! (n 1)! In the following we will introduce the Laguerre polynomial family by analogy with (4)(1), where n ( )n, n!. (46) ˆD L ˆD Thus, we find the epression for the Laguerre polynomials l n () =A( L ˆD ) [ ] ( ) n n! (47) and in the same way we can define the complementary Laguerre polynomials ln() = [ A( L ˆD ) ] [ ] 1 ( ) n. (48) n! 3 The two-variable Hermite polynomials H n (, y) =(i y) n H n ( [n/2] ( 1) Hermite polynomials H n () =n! r (2) n 2r y r (n 2r)!r!. 2i y ) contain ordinary
Appèl polynomial series epansions 657 Therefore, drawing an analogy with the Appèl polynomials we can obtain a similar epansion in series, e. g.: f() = α n l n (), α n = ( 1) r f (r) (0) β r,n, β r,n = ( 1)n 2π 2π 0 e inφ a r(e iφ )dφ. (49) The Laguerre polynomials defined in (43) are in fact Appèl polynomials with respect to the y variable and they can be eplicitly constructed according to the following operational rule: L n (, y) =C 0 ( ˆD y )y n, C n () = ( 1) r r = r!(r+n)! n 2 J n (2 ). (50) Accordingly, we can derive the complementary Laguerre polynomials L 1 n(, y) = C 0 ( ˆD y ) yn. (51) However, their derivation is not straightforward, being the epansion of the 0-th order Tricomi function, given by [C 0 ()] 1 =1+ + 3 4 2 + 19 36 3 + 211 576 4 + o( 5 ) (52) For the first few polynomials we find the following epressions: L 0 (, y) =1,L 1 (, y) = + y, L 2 (, y) =y2 +2y + 3 2 2, L 3(, y) =y 3 +3y 2 + 9 2 (y)2 + 19 6 3, (53) L 4 (, y) =y4 +4y 3 +9(y) 2 + 19 6 y3 + 211 24 4 The polynomials defined by the Appèl operator C 0 ( ˆD y 2 ) belong to the so called hybrid family, defined by the following sum: [n/2] P n (, y) =n! ( 1) r r y n 2r (r!) 2 (n 2r)! (54) and their name arises from their properties, laying somewhere between those of Laguerre and Hermite polynomials. However, they are quite interesting. For eample, for =1 y 2 /4 they reduce to the Legendre polynomials. The epansion procedure for them is the same as described above for other polynomial families, but we do not dwell on this aspect of the problem for the
658 G. Dattoli and K. Zhukovsky sake of conciseness. Further comments on the results obtained in this section will be given in the concluding remarks, where we will also treat the case of d orthogonal polynomials. 4 Multi-inde Appèl polynomials and concluding remarks We define multi-inde (and multi-variable) Appèl polynomials 4 a m,n (, y) =A( ˆD ˆDy )( m y n ), (55) where the operator A depends on the product of the derivative operators and it is supposed to be such that the Inverse Appèl operator (5) eists. The eample of the two-inde polynomials, belonging to the Appèl family, is provided by the incomplete Hermite polynomials, (see [7], [8]) h m,n (, y τ) = ep(τ ˆD [m,n] ˆDy )( m y n τ r m r y n r )=m!n! r!(m r)!(n r)!. (56) It should be noted that, in general, we have two distinct derivative operators: ˆD a m,n (, y) =ma m 1,n (, y), ˆD y a m,n (, y) =na m,n 1 (, y) (57) and two multiplicative operators 5 ˆM = + ˆD A y, A ˆM y = y + ˆD A [, A ˆM, ˆM ] y = ˆM ˆMy ˆM y ˆM =0, (58) so that ˆM a m,n (, y) =a m+1,n (, y), ˆM y a m,n (, y) =a m,n+1 (, y). (59) Provided the inverse operator eists, we define the correspondent associated polynomials a m,n(, y) = [ A( ˆD ˆDy ) ] 1 ( m y n ). (60) 4 We are defining the case of two indices only. The etension to more than two indices is straightforward. 5 The commutativity between the multiplicative operators, ensured by the last of the eqs. (54), is essential for what follows.
Appèl polynomial series epansions 659 Now we can generalize the above obtained results and derive the epansion of a given two-variable function with respect to a m,n as follows: f(, y) =α m,n a m,n (, y), α m,n = f (r,s) β r!s! r,s,m,n, r,s=0 β r,s,m,n = 1 (2π) 2 2π 0 dφ 2π 0 dτa m,n (eiφ,e iτ )e i(mφ+nτ). (61) We will continue the discussion of the problem of the multi-inde Appèl polynomials in forthcoming works. Moreover, we can demonstrate that the method discussed in the present article and illustrated on the eamples of various polynomial families is amenable to further generalizations. Indeed, let us consider the polynomials r n () generated by the Appèl operator in the following way: A(t)f(t) = t n n! r n(), (62) where f() is not an eponential, but it admits the series epansion f() = a n n! n. Then, we can define the polynomials r n () =A ( f ˆD ) n, (63) where f ˆD is the umbral derivative or the Δ operator and it is defined in the way, such that: f ˆD n = na n 1 n 1 (64) Thus, in the case of A(t) = ep(t 2 ) we obtain which reduces for f() =C m () = [n/2] a n 2r n 2r r n () =n! r!(n 2r)!, (65) ( 1) r r r!(n+r)! to [n/2] n 2r r n () =n! r!(n 2r)!(m + n 2r)!, (66) belonging to the family of the Hybrid polynomials, discussed in [9]. The method of the series epansion can be easily etended to include also the family of hybrid polynomials. More general eamples in terms of Boas- Buck and Sheffer polynomials will be presented in forthcoming publications.
660 G. Dattoli and K. Zhukovsky It is worth to emphasize with reference to the above remark, that there are polynomials, which cannot be recognized as Appèl. An eample of the above said is provided by the polynomials u n () = n k=0 k (n k)! (67) with the generating function t n u n () = et 1 t. (68) Even though they are not the Appèl type polynomials, they can be easily epressed in terms of the Appèl polynomials by noting that u n () = n e n ( 1 ) (69) It is also interesting to note that if we perform in the eq. (68) the following substitution: ˆM 1 =(1 ˆD ) (70) ˆD 1 with the inverse of the derivative operator and ˆM the multiplicative operator of the ordinary Laguerre polynomials [9], then we find 6 t n π n () = et e t 1 t 1 t, (71) so that we can interpret π n () as a Sheffer type polynomial. In particular, we obtain: n L k () π n () = (n k)!, (72) k=0 which can be recognized as the two-orthogonal Douhat-Laguerre type function. Concluding our studies of the Appèl polynomials we would like demonstrate the usefulness of the above obtained results on the eample of the heat equation 2 F (, t) = F (, t), t 2 F (, 0) = f(). (73) 6 In order to derive (67) we used the property refs [9]). n ˆD m m! = n+m (n+m)! (for further details see
Appèl polynomial series epansions 661 On the assumption of the eistence of the epansion of the initial function with respect to Hermite polynomials H n (, y) we can write the solution of our Cauchy problem as follows: F (, t y ) = ep(t 2 ) α 2 n H n (, y) = α n H n (, y + t), (74) so that the relevant solution is obtained in terms of a simple shift of the y variable. In forthcoming investigations we will provide further eamples, including the etension to higher order heat equations. ACKNOWLEDGMENTS The author epresses his sincere appreciation to Profs. P.E. Ricci, B. Germano and M.R. Martinelli for checking the manuscript and for a number of enlightening suggestions. REFERENCES 1. H. M. Srivastava and H. L. Manocha, A Treatise on generating functions Ellis-Horwood series: Mathematics and its applications, John Wiley & Sons, New York (1984). 2. For the conditions invertibility of the Appèl operator see M. Craciun Approimation methods obtained using the umbral calculus Doctoral Thesis BABE S- Bolyai University of CLUJ-NAPOCA Faculty of Mathematics and Computer Science. 1. G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions a by-product of the monomiality principle Advanced special functions and applications (Melfi, 1999) (D. Cocolicchio, G. Dattoli and H. M. Srivastava eds.) Proc. Melfi Sch. Adv. Top. Math., Vol. 1, Rome 2000, pp. 147-164. 2. L. C. Andrews, Special functions for engineers and applied mathematicians Mc. Millan, New York (1985). 3. F. Gori, Flattened Gaussian beams, Opt. Commun. 107, 1994, pp. 335-341.
662 G. Dattoli and K. Zhukovsky 4. G. Dattoli and M. Migliorati, Int. J. Math. & Math Sci. Vol. 2006, article ID 98175, pag.1-10, Doi 10.1155/IJMMS/2006/98175 5. A. Wünsche, Laguerre 2-D functions and their application in quantum optics. J. Phys. A 31 (1999), pp. 267 270. 6. G. Dattoli. Incomplete 2D Hermite polynomials: properties and applications Journal of Mathematical Analysis and Applications. Volume 284, Issue 2, 15 August 2003, Pages 447-454. 7. K. Douak, Int. J. Math. & Math. Sci., 22, 29 (1999). 8. G. Dattoli, B. Germano and P. E. Ricci Comments on monomiality, ordinary polynomials and associated bi-orthogonal functions. Applied Mathematics and Computation Volume 154, Issue 1, 25 June 2004, Pages 219-227. Received: March, 2009