App`el Polynomial Series Expansions - CiteSeerX

International Mathematical Forum, 5, 2010, no. 14, 649 - 662

App` el Polynomial Series Expansions G. Dattoli ENEA UTS Fisiche Avanzate Centro Ricerche Frascati C.P. 67 - 00044 Frascati, Rome, Italy [email protected] K. Zhukovsky Dept. of Optics and Spectroscopy, Faculty of Physics M. V. Lomonosov Moscow State University Moscow 119991, Russia [email protected] 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 exploited to derive general criteria to get the expansion of a given function in terms of the above polynomial families. We discuss the case of multi-index Appèl polynomials and touch on the Sheffer families.

Keywords: Appél polynomials, Sheffer polynomials, series expansion, orthogonal polynomials, multi-index polynomials 1. Introduction The Appèl polynomials an (x) are defined through the following generating function [1]: ∞ n t n=0 n!

an (x) = A(t)ext ,

(1)

where A(t) is left, at the moment, unspecified, but it is assumed that there exists a finite region of t, in which the expansion of A(t)in Taylor series converges.

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G. Dattoli and K. Zhukovsky

The use of the obvious identity ˆ x ext , text = D ˆx = d D dx

(2)

and the previous assumption that A(t) can be expanded in Taylor series, allows to recast eq. (1) in the following operational form: ∞ n t n=0 n!

ˆ x )ext . an (x) = A(D

(3)

Therefore, expanding the exponential 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 an (x):

ˆ x xn . an (x) = A D

(4)

ˆ x as to the Appèl operator and we will In what follows we will refer to A D −1 ˆx assume that, along with it the inverse of the Appèl operator A D (see [2]) can be defined in such a way that

ˆ x) A(D

−1

ˆ x ) = ˆ1. A(D

(5)

Note, for example, that in the case of the two-variable Hermite polynomials the identity (4) specializes as follows [3]: ˆ2

Hn (x, y) = eyDx xn , Hn (x, y) = n!

[n/2] r=0

xn−2r y r (n−2r)!r!

,

(6)

while for the Truncated Exponential Polynomials (TEP) we get [3] e¯n (x) = e¯n (x) =

1 n ˆx x , 1−D n xr n! r! r=0

(7)

The operational definition (4) can now be exploited to derive the main properties of the Appèl polynomials. Indeed, by taking the derivative of both sides ˆx of the eq. (4) with respect to x, we obtain, on account of the operators A D ˆ x commute1 : and D ˆ x ), D ˆx = 0 A(D (8) ˆ B ˆ = AˆB ˆ −B ˆ Aa ˆ commutation bracket between the two We have denoted by A, operatorsA, B. 1

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Appèl polynomial series expansions

the well known property ˆ x an (x) = nan−1 (x). D

(9)

Furthermore, from the (4) we write ˆ x )x]xn an+1 (x) = [A(D

(10)

ˆ and then, according to eq. (4) we can introduce the multiplicative operator M (see [2]) in the following way: ˆ an (x) an+1 = M

(11)

ˆ specified in terms of the Appèl operator as given below: Thus, with M

ˆ x xA D ˆx ˆ =A D M

−1

,

(12)

the usage of the identity

ˆ x ), x = f (D ˆ x ), f (D

(13)

where f is the derivative of the function f , yields:

ˆ = x + A(D ˆ x) M

−1

ˆ x ). A (D

(14)

ˆ and D ˆ x realize the multiplicative and derivative operators The operators M for the Appèl polynomial family, which, therefore can be viewed as the family ˆ D ˆ and ˆ1 of quasi monomials. It is also evident that the set of operators M, realizes the Weyl-Heisenberg algebra. Furthermore, by noting that ˆD ˆ x an (x) = nan (x) M

(15)

we find that the Appèl polynomials satisfy the following equation, involving ˆ x , which we will call the differential equation for the derivation operator D Appèl polynomials: ˆ x an (x) + xD

ˆ x) A (D ˆ a (x) = nan (x), D ˆ x) x n A(D

(16)

where A is the derivative of A. This equation is valid for all of the polynomials, belonging to the Appèl family.

652


In the particular case of the Hermite polynomials we obtain the multiplicaˆ tive operator Mand the differential equation for the Appèl polynomials as follows: ˆ = x + 2y D ˆx M (17) ˆ 2 Hn (x, y) + xD ˆ x Hn (x, y) = nHn (x, y) , 2y D x whereas for the Truncated Exponential Polynomials we obtain: ˆ =x+ 1 , M ˆx 1−D . 2 ˆ ˆ x e¯n (x) + n¯ xDx e¯n (x) − (x + n)D en (x) = 0

(18)

Further examples 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 ˆ x) A (D (19) )a (x). an+1 (x) = (x + ˆ x) n A(D Being orthogonal polynomials, they are also characterized by the identity on+1 (x) = (an + bn x)on (x) + cn on−1 (x)

(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 expansions in terms of the an (x) polynomials can be defined in general terms, without any explicit use of their orthogonal nature. 1. The Complementary App` el polynomials and series expansion Let us define the family of polynomials a∗n (x):

ˆ x) a∗n (x) = A(D

−1

xn

(21)

through the inverse of the Appèl operator. Let us call this family the an (x)– complementary. They belong to the Appèl family and they satisfy the following recurrences: ˆ a∗ (x) = na∗n−1 (x), D x n . (22) ˆ x) A (D x − A(Dˆ ) a∗n (x) = a∗n+1 (x) x

653


It is easily checked that the complementary Hermite polynomials are expressed in terms of the ordinary Hermite polynomials simply as Hn (x, −y) while the complementary TEP are e¯∗n (x) = xn − nxn−1 . Let us assume that a given functionf (x), admitting a Taylor expansion around the origin, can be expanded in Appèl series too, namely: ∞

f (x) =

αn an (x).

(23)

n=0

Our goal is now to determine the expansion coefficients αn . According to eq. (4) we can write the expansion (23) in the form2 ˆ x) f (x) = A(D

∞

αn xn ,

(24)

n=0

which can be inverted according to the following rule:

ˆ x) A(D

−1

f (x) =

∞

αn xn

(25)

n=0

and from the common Taylor expansion of the function f (x) = can conclude by exploiting the eq. (21) that ∞

∞ f (r) (0) ∗ ar (x) = αn xn . r! r=0 n=0

∞ (n) f (0) n x

n=0

n!

we

(26)

The use of the orthogonality of the circular functions allows the derivation of the coefficients αn from the identity αn =

∞ (r) f (0) r=0

βr,n =

1 2π

r! 2π

0

βr,n ,

einϕ a∗r (eiφ )dφ

.

(27)

In the case of the Hermite polynomials we find: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

βr,n =

2

The identity

∞ n=0

⎪ ⎪ ⎪ ⎪ ⎩

r − n ≡ odd

0; r−n

(28)

(−y) 2 ;r r! n! ! ( r−n 2 )

ˆ x )xn = A(D ˆ x) αn A(D

∞ n=0

− n ≡ even,

αn xn is justified by the assumption that the

Appèl operator can be expanded in Taylor series and by the consequent assumption that ∞ m n! αn xn and αn (n−m)! xn−m converge uniformly. the series n=0

n=0

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A cylindrical Bessel function can be expanded in the following series of the Hermite polynomials with the coefficients αn : Jm (x) = αn =

m ∞ x 2

n=0 ∞ (−1)r βr,n r=n (m+r)! r!

αn Hn

x2 ,y 4

,

,

(29)

where β r,n are given by (28). For the expansion of a cylindrical Bessel function in terms of TEP polynomials we find for the coefficients β r,n βn,r = δn,r − rδn,r−1

(30)

and therefore the expansion of the Jn (x) in TEPs reads as follows: n ∞ (−1)s (n+s+2) x

Jn (x) = en (x) =

2

n xr r=0

s=0

(n+s+1)!

es

x2 4

.

(31)

r!

Now let us consider the Euler polynomials En (x), which are defined by the generating function [1], [4]: ∞ n t n=0

n!

en (x) =

2ext , |t| < π. et + 1

(32)

Then the relevant Appèl operator and its inverse have the following form: ˆ x) = ˆ 2 , A(D eDx +1 . ˆx D −1 ˆ A(Dx ) = e +1

(33)

2

We also obtain substituting the above expression for the Appèl operator into (21) the Complementary Euler polynomials e∗n (x) =

1 [(x + 1)n + xn ] . 2

(34)

Thus, we write the expansion of a given function in series of the Euler polynomials: ∞ f (x) = αn en (x), n=0

αn =

∞

f (r) (0) r! r=n

⎡⎛

⎞

⎤

. r ⎠ ⎣⎝ + 1⎦ n

(35)

The TEP are often exploited in optics to treat the propagation of the so called flattened beams [5]; recently the present author has exploited higher order

655


TEP to study the propagation of beams having the particular configuration – a transverse distribution with a hole [6]. The higher orders TEP: e¯n (x|m) =

[n/m] xn−mr 1 xn = n! ˆm 1−D r=0 (n − mr)! x

(36)

are Appèl polynomials too. The associated complementary forms are e¯∗n (n|m) = xn −

n! xn−m . (n − m)!

(37)

For reasons we will clarify in what follows, we are mainly interested in expressing the e¯n (x|m) in terms of the higher order Hermite polynomials, which can be defined by the following operational identity: Hn(m) (x, y)

ˆ xm )xn = n! = exp(y D

[n/m]

r=0

xn−mr y r . (n − mr)!r!

(38)

With the help of the Laplace transform identity we write : 1 = ˆm 1−D x

∞

ˆm

e−s esDx ds

(39)

0

and then we derive accounting for (36) and (38) the following integral form for the TEP: ∞

e¯n (x|m) =

e−s Hn(m) (x, s)ds.

(40)

0

The above formula (40), along with the definition of the Hermite polynomials (38) suggest that ˆm

eτ Dx Hn(m) (x, y) = Hn(m) (x, y + τ ).

(41)

Thus the “propagation” property of the higher order TEP follows: ˆm τD x

e

∞

e¯n (x|m) =

e−s Hn(m) (x, s + τ )ds.

(42)

0

We will comment on the usefulness of the above expansions in the concluding section. 3 App` el and Laguerre Polynomials

656


The introduction of an extra 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 transformation3 , 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 context 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 Ln (x, y) = n!

n

(−1)r y n−r xr 2 r=0 (n − r)! (r!)

(43)

reduce to ordinary Laguerre polynomials for y = 1 are, indeed, not Appèl polynomials with respect to x. The two variable Laguerre polynomials can indeed defined in terms of the following operational identity: ˆ

Ln (x, y) = eyL Dx ˆ ˆ ˆ L D x = −Dx xDx

(−x)n n!

,

.

(44)

ˆx It can be easily checked by noting the action of the Laguerre derivative L D n on (−x) : n! (−x)n (−x)n−1 ˆ =n . (45) L Dx n! (n − 1)! In the following we will introduce the Laguerre polynomial family by analogy with (4)(1), where n , xn → (−x) n! (46) ˆ x → LD ˆx . D Thus, we find the expression for the Laguerre polynomials

n ˆ x ) (−x) ln (x) = A(L D n!

(47)

and in the same way we can define the complementary Laguerre polynomials ln∗ (x) 3

−1

ˆ x) = A(L D

(−x)n . n!

(48)

√ The two-variable Hermite polynomials Hn (x, y) = (i y)n Hn ( 2ix√y ) contain ordinary

Hermite polynomials Hn (x) = n!

[n/2] r=0

(−1)r (2x)n−2r y r . (n−2r)!r!

657


Therefore, drawing an analogy with the Appèl polynomials we can obtain a similar expansion in series, e. g.: f (x) = α ñ =

∞

n=0 ∞

r=0

β˜r,n =

α ˜ n ln (x),

(−1)r f (r) (0)β˜r,n ,

(−1)n 2π

2π

0

.

(49)

einφ a∗r (eiφ )dφ

The Laguerre polynomials defined in (43) are in fact Appèl polynomials with respect to the y variable and they can be explicitly constructed according to the following operational rule: ˆ y )y n , Ln (x, y) = C0 (xD Cn (x) =

∞ (−1)r xr

r=0

r!(r+n)!

√ . n = x− 2 Jn (2 x)

(50)

Accordingly, we can derive the complementary Laguerre polynomials L∗n (x, y) =

1 y n. ˆ C0 (xDy )

(51)

However, their derivation is not straightforward, being the expansion of the 0-th order Tricomi function, given by 3 19 211 4 x + o(x5 ) [C0 (x)]−1 = 1 + x + x2 + x3 + 4 36 576

(52)

For the first few polynomials we find the following expressions: L∗0 (x, y) = 1, L∗1 (x, y) = x + y, L∗2 (x, y) = y 2 + 2xy + 32 x2 , L∗3 (x, y) = y 3 + 3xy 2 + 92 (xy)2 + 19 x3 , 6 yx3 + 211 x4 L∗4 (x, y) = y 4 + 4xy 3 + 9(xy)2 + 19 6 24

(53)

ˆ 2 ) belong to the so called The polynomials defined by the Appèl operator C0 (xD y hybrid family, defined by the following sum: [n/2]

Pn (x, y) = n!

r=0

(−1)r xr 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 example, for x = 1 − y 2 /4 they reduce to the Legendre polynomials. The expansion 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


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-index App` el polynomials and concluding remarks We define multi-index (and multi-variable) Appèl polynomials4 ˆ xD ˆ y )(xm y n ), am,n (x, y) = A(D

(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) exists. The example of the two-index polynomials, belonging to the Appèl family, is provided by the incomplete Hermite polynomials, (see [7], [8]) ˆ y )(xm y n ) = m!n! ˆ xD hm,n (x, y|τ ) = exp(τ D

[m,n]

r=0

τ r xm−r y n−r . r!(m − r)!(n − r)!

(56)

It should be noted that, in general, we have two distinct derivative operators: ˆ x am,n (x, y) = mam−1,n (x, y), D ˆ y am,n (x, y) = nam,n−1 (x, y) D

(57)

and two multiplicative operators5 ˆx = x + D ˆ y A , M A ˆy = y + D ˆ x A , M , A ˆ ˆ ˆ ˆ ˆ ˆ Mx , My = Mx My − My Mx = 0

(58)

ˆ x am,n (x, y) = am+1,n (x, y), M ˆ y am,n (x, y) = am,n+1 (x, y) . M

(59)

so that

Provided the inverse operator exists, we define the correspondent associated polynomials ˆ xD ˆ y ) −1 (xm y n ). (60) a∗m,n (x, y) = A(D 4

We are defining the case of two indices only. The extension 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.

659


Now we can generalize the above obtained results and derive the expansion of a given two-variable function with respect to am,n as follows: f (x, y) = αm,n am,n (x, y), αm,n =

∞ f (r,s)

r,s=0

βr,s,m,n =

r!s!

1 (2π)2

βr,s,m,n ,

2π

0

dφ

2π

0

.

(61)

dτ a∗m,n (eiφ , eiτ )e−i(mφ+nτ )

We will continue the discussion of the problem of the multi-index Appèl polynomials in forthcoming works. Moreover, we can demonstrate that the method discussed in the present article and illustrated on the examples of various polynomial families is amenable to further generalizations. Indeed, let us consider the polynomials rn (x) generated by the Appèl operator in the following way: ∞ n t

A(t)f (xt) =

n=0

n!

rn (x),

(62)

where f (x) is not an exponential, but it admits the series expansion f (x) = ∞ an n x .

n=0

n!

Then, we can define the polynomials rn (x) = A

ˆ

f Dx

xn ,

(63)

ˆ x is the umbral derivative or the Δ operator and it is defined in the where f D way, such that: n−1 ˆ n (64) f Dx x = nan−1 x Thus, in the case of A(t) = exp(t2 ) we obtain [n/2]

rn (x) = n!

r=0

which reduces for f (x) = Cm (x) = [n/2]

rn (x) = n!

r=0

an−2r xn−2r , r!(n − 2r)!

∞ (−1)r xr r=0

r!(n+r)!

(65)

to

xn−2r , r!(n − 2r)!(m + n − 2r)!

(66)

belonging to the family of the Hybrid polynomials, discussed in [9]. The method of the series expansion can be easily extended to include also the family of hybrid polynomials. More general examples in terms of BoasBuck and Sheffer polynomials will be presented in forthcoming publications.

660


It is worth to emphasize with reference to the above remark, that there are polynomials, which cannot be recognized as Appèl. An example of the above said is provided by the polynomials un (x) =

n

xk k=0 (n − k)!

(67)

et . 1 − tx

(68)

with the generating function ∞

tn un (x) =

n=0

Even though they are not the Appèl type polynomials, they can be easily expressed in terms of the Appèl polynomials by noting that

un (x) = xn en

1 x

(69)

It is also interesting to note that if we perform in the eq. (68) the following substitution: ˆ = (1 − D ˆ −1 ) (70) x→M x ˆ – the multiplicative ˆ −1 – the inverse of the derivative operator and M with D x operator of the ordinary Laguerre polynomials [9], then we find6 tx

et e− 1−t , t πn (x) = 1−t n=0 ∞

n

(71)

so that we can interpret πn (x) as a Sheffer type polynomial. In particular, we obtain: n Lk (x) , (72) πn (x) = (n − k)! 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 example of the heat equation ∂ F (x, t) ∂t

2

∂ = ∂x 2 F (x, t), . F (x, 0) = f (x)

ˆ −n xm = In order to derive (67) we used the property D x m! refs [9]). 6

(73) xn+m (n+m)! (for

further details see


661

On the assumption of the existence of the expansion of the initial function with respect to Hermite polynomials Hn (x, y) we can write the solution of our Cauchy problem as follows: F (x, t |y ) = exp(t

∞ ∞ ∂2 ) α H (x, y) = αn Hn (x, y + t), n n ∂x2 n=0 n=0

(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 examples, including the extension to higher order heat equations. ACKNOWLEDGMENTS The author expresses 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 “Approximation 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.

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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. W¨ unsche, 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