Monomiality and partial differential equations

Mathematical and Computer Modelling 50 (2009) 1332–1337

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Monomiality and partial differential equations G. Dattoli a , B. Germano b , M.R. Martinelli b , P.E. Ricci c,∗ a

Unità Tecnico Scientifica Tecnologie Fisiche Avanzate, ENEA – Centro Ricerche Frascati – C.P. 65, Via E. Fermi, 45, 00044 – Frascati – Roma, Italy

b

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma ‘‘La Sapienza’’, Via A. Scarpa, 14, 00161 – Roma, Italy

c

Dipartimento di Matematica ‘‘Guido Castelnuovo’’, Università degli Studi di Roma ‘‘La Sapienza’’, P.le A. Moro, 2, 00185 – Roma, Italy

article

info

Article history: Received 25 June 2007 Received in revised form 4 June 2009 Accepted 5 June 2009 Keywords: Monomiality principle Appell polynomials Sheffer polynomials Partial differential equations Integro-differential equations Initial value problems

abstract We show that the combination of the formalism underlying the principle of monomiality and of methods of an algebraic nature allows the solution of different families of partial differential equations. Here we use different realizations of the Heisenberg–Weyl algebra and show that a Sheffer type realization leads to the extension of the method to finite difference and integro-differential equations. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction

b satisfying the rules of commutation Since the foundations of Quantum Mechanics, the nilpotent algebra of b P, M b ] = 1, [b P, M

b , 1] = 0 [b P , 1] = [M

(1)

has been the natural environment to treat problems associated with the canonical quantization (see [1]). Since then the Weyl–Heisenberg algebra has been exploited in other problems ranging from quantum optics [2] to string theory for some discrete models of two-dimensional theories [3]. The generators of the nilpotent algebra can be realized in many different ways and non standard realizations have stimulated the development of the monomiality principle [4,5]. We recall that a given family of polynomials pn (x) is said to be ‘‘quasi-monomial’’ under the action of two operators b P b called ‘‘derivative’’ and ‘‘multiplicative’’ operators respectively, if it satisfies the recurrences and M,

b n (x) = pn+1 (x) Mp b Ppn (x) = npn−1

(2a)

pn (0) = 1.

b verify the commutation property The b P and M b] = b b−M bb [b P, M P M P = 1,

(2b)

and thus display a Weyl group structure.

∗

Corresponding author. E-mail addresses: [email protected] (G. Dattoli), [email protected] (B. Germano), [email protected] (M.R. Martinelli), [email protected], [email protected] (P.E. Ricci). 0895-7177/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2009.06.013

G. Dattoli et al. / Mathematical and Computer Modelling 50 (2009) 1332–1337

1333

b have a differential realization, then the polynomial pn (x) satisfies the differential equation Furthermore if b P and M bb M P (pn (x)) = npn (x),

(3)

pn (x) can be constructed as

b n (1) pn (x) = M

(4)

assuming p0 (x) = 1 and the exponential generating function of pn (x) can be put in the form et M (1) = b

∞ n X t n=0

n!

pn (x).

(5)

The interest for monomiality and for the associated techniques is manifold. Its most appealing feature is indeed that of providing the simultaneous solution of a number of apparently different problems by the use of a very simple formalism employing different realizations of the multiplicative and derivative operators. According to this point of view we can obtain the solution of a large class of families of ordinary or partial differential problems by combining the momomiality techniques with the more conventional methods like eigenvalue or evolution operator methods. In this paper we consider the realization of the derivative and multiplication operators for the Appell and Sheffer polynomials and discuss the wealth of consequences one can obtain for the explicit solution of either ordinary or partial differential equations. 2. Monomiality and Appell polynomials The Appell polynomials, specified by the generating function, ∞ n X t n =0

n!

an (x) = A(t ) exp(xt )

(6)

where A(t ) is a continuous function with at least one derivative, can be shown to be quasi-monomials using the procedure suggested in Ref. [6]. By keeping the derivative of both sides of (6) with respect to x, we get ∞ n X t n =0

n!

b Dx an (x) = tA(t ) exp(xt ) =

∞ n +1 X t n=0

n!

an (x),

(7)

d b Dx = .

dx By rescaling the index and comparing the t-like powers, we find

b Dx an (x) = nan−1 (x) thus implying that the b P operator of the Appell is just the ordinary derivative. Eq. (6) can also be written in the form ∞ n X t

an (x) = A(b Dx ) exp(xt ) = A(b Dx )

∞ n X t

xn n ! n ! n =0 n =0 which allows the conclusion that the Appell polynomials can be defined through the operational rule an (x) = A(b Dx )xn .

(8)

(9)

(10)

The multiplicative operator can now be explicitly defined by noting that an+1 (x) = A(b Dx )[x(xn )] = A(b Dx )x[A(b Dx )]−1 an (x),

b = A(b M Dx )x[A(b Dx )]−1 = x + [A(b Dx ), x][A(b Dx )]−1 = x + [A(Dx )]−1 A0 (Dx )

(11)

where the prime denotes the derivative of the function A(t ). According to the previous results and the identity given in Eq. (3), it can easily be shown that the Appell polynomials satisfy the differential equation

" x+

A0 (b Dx )

#

A(b Dx )

b Dx an (x) = nan (x).

(12)

It is well known that many ‘‘popular’’ polynomials belong to the Appell family, we can therefore use the general equation (12) to recover all very well known particular cases. The Hermite polynomials belong to the Appell family and they are indeed characterized by the function



A(t ) = exp −

t2 2



.

(13)

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We immediately recover the differential equation they satisfy as d [x − b Dx ] Hen (x) = nHen (x).

(14)

dx

If for example A(t ) = t /(et − 1), the associated Appell can be recognized as Bernoulli polynomials and the underlying Weyl–Heisenberg operators are realized by

" # b (1 − b Dx )eDx − 1 b M = x+ , (ebDx − 1)b Dx

b P =b Dx .

(15)

By applying Eq. (12) we find that the equation they satisfy is no longer an ODE, but it can be written in the form of a differential-difference equation. These two examples are paradigmatic of what we intend to do in this article, namely to take advantage from the theory of monomiality to discuss classes of exactly solvable differential equations. 3. Monomiality and Sheffer polynomials The Sheffer polynomials are the natural extension of the Appell polynomials, they are indeed generated by ∞ n X t n =0

n!

sn (x) = A(t ) exp(xB(t )),

(16)

with B(t ), satisfying the same properties as A(t ). We follow the point of view of Ref. [6] to prove their quasi-monomiality. To this aim we take the freedom of calling Pre-Sheffer the polynomials s˜n (x), with the generating function ∞ n X t n =0

n!

s˜n (x) = exp(xB(t )).

(17)

Multiplying both sides of Eq. (17) by the operator B−1 (b Dx ), we find that ∞ n X t

n! n =0

B−1 (Dx )˜sn (x) = B−1 (Dx ) exp(xB(t )) = t exp(xB(t )),

(18)

thus finding that the derivative operator for this family of polynomials is

b P = B−1 (b Dx )

(19)

the multiplicative operator can now be defined by taking the derivative with respect to t of both sides of Eq. (14) thus finding ∞ X nt n−1 n =0

n!

s˜n (x) = xB0 (t ) exp(xB(t )) = xB0 (B−1 (b Dx )) exp(xB(t )),

(20)

which implies that the relevant multiplicative operator can be cast in the form

b = xB0 (b M P)

(21)

with b P given by Eq. (19). Let us now consider the Sheffer polynomials and note that they can be defined through the operational rule sn (x) = A(b P )˜sn (x).

(22)

It is now clear that the Sheffer polynomials are the corresponding Appell on the space of the Pre-Sheffer polynomials, according to the following identification xn ↔ s˜n (x),

(23)

b Dx ↔ B−1 (b Dx ), thus finding for the Sheffer polynomials the following realization of the Weyl–Heisenberg operators

b P = B−1 (Dx ), b = A(b M P )xB0 (b P )A(b P ) −1

(24)

being [A(b P ), xB0 (b P )] = A0 (b P ) we can rewrite the multiplicative operator as

b = xB0 (b M P) +

A0 (b P) A(b P)

.

(25)

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The Sheffer polynomials therefore satisfy the differential equation

"

A0 (b P)

xB0 (b P) +

# b Psn (x) = nsn (x).

A(b P)

(26)

We have now a general and direct way to generate families of non trivial realization of the nilpotent algebra and explore forms of isospectral problems and the exact solution of partial differential equations. An interesting example is provided by the pre-Sheffer polynomials s˜n (x) linked to the Bessel polynomials by s˜n (x) = xn yn−1

  1 x

,

(27)

 n   X n n+k

y n ( x) =

k

k=0

k

k!

 x k 2

where yn (x) denotes the Bessel polynomials introduced by Krall and Frinck. The above polynomials belong to the quasi-monomial family and the relevant characteristic operators can be realized as follows Ref. [6] 1

b=x M

,

1 −b Dx (28) 1 _2 b P = − Dx +b Dx . 2 We will see in the following that the above realization allows interesting and useful conclusions either from the purely mathematical and applicative side. The use of the identity A

−1

∞

Z

esA ds

=

(29)

0

allows us to rewrite the multiplicative operator in the form ∞

Z

exp(−s) exp(sb Dx ) ds

b=x M

(30)

0

and, since exp(λb Dx )f (x) = f (x + λ), we can easily conclude that its action on a given function is specified by the integral transform

b (x) = x Mq

∞

Z

exp(−s)q(x + s) ds,

(31)

0

valid whenever the integral on the r.h.s. of Eq. (31) converges. In the following sub-sections we will discuss examples which can be solved by means of the above developed formalism. 3.1. Non local ‘‘evolution’’ integro-differential problem

  Z ∞ ∂ 1 2 F (x, y) = − Dx + Dx F (x, y) + x e−s F (x + s, y) ds, ∂y 2 0 F (x, 0) = q(x).

(32)

The above equation can be written in the form

∂ b )F (x, y), F (x, y) = (b P +M ∂y F (x, 0) = q(x),

(33)

so that its formal solution writes,

b +b F (x, y) = exp(y(M P ))q(x).

(34)

The use of the Weyl decoupling rule b b b b eA+B = e−k/2 eA eB ,

if [b A, b B] = k, k ∈ C

yields 1 2 b b b b F (x, y) = ey(P +M ) q(x) = e− 2 y eyP eyM q(x).

(35)

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The explicit action of the above exponential operators can be also easily obtained using standard integral transforms or other means, thus getting e.g. for the exponential containing the multiplicative operator eyM f (x) = b

∞ X yn

n!

n =0

πn (x) = x

πn (x),

(36)

∞

Z

e−s πn−1 (x + s) ds. 0

3.2. Initial value problem Let us now consider the initial value problem

  1 _ ∂ − D2y +b R(y, τ ), Dy R(y, τ ) = 2 ∂τ R(0, τ ) = a(τ )

(37)

which can formally be written as

∂ b R(y, τ ), PR(y, τ ) = (38) ∂τ R(0, τ ) = a(τ ), where b P is the derivative operator associated with the polynomials s˜n (x) specified in Eq. (27). It is evident that we can now obtain the solution of the previous problem using the same procedure described in Section 3.1, if we are able to derive the eigenfunction of the operator b P, which in the case of the realization (28) is shown to be (see also Ref. [6])

h

f (x, λ) = exp x(1 −

√

i

1 − 2λ) =

∞ X λn n =0

n!

sn (x).

(39)

It is indeed easily checked that

b Pf (x, λ) = λf (x, λ).

(40)

We can therefore write the solution of the ‘‘initial value problem’’ (38) as

" R(y, τ ) = exp y 1 −

r

∂ 1−2 ∂τ

#! a(τ ),

(41)

using Eq. (38) and expanding in terms of the sn (x) polynomials, we end up with R(y, τ ) =

∞ X 1 sn (y)a(n) (τ ) n ! n=0

(42)

where a(n) (τ ) means the n-th derivative with respect to τ . The validity of this last result is limited to cases in which the series on the l.h.s. of Eq. (42) converges. 4. Concluding remarks In Section 3 we have seen how the use of simple prescriptions of algebraic nature allows the solution of partial differential equations whose solution could hardly be achieved using conventional means. In this section we will discuss examples involving different type of Sheffer polynomials, leading to the solution of different families of PDE of different nature. The Bell polynomials are specified by the generating function ∞ n X t n =0

n!

Ben (x) = exp(x(et − 1))

(43)

(see [4]) which, according to the discussion developed in Ref. [6], can be written in terms of the Stirling numbers of the second kind as follows Ben (x) =

n X

S2 (n, k)xk

(44a)

k=1

or in terms of the operational definition Ben (x) = exp(−x)(xb Dx )n exp(x).

(44b)

G. Dattoli et al. / Mathematical and Computer Modelling 50 (2009) 1332–1337

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The monomiality operator associated with the above polynomials are

b = x(1 + b M Dx ),

(45)

b P = ln(1 + b Dx ). The solution of an equation of the type ln(1 + b Dy )R(y, τ ) = R(0, τ ) = a(τ ),

∂ (y, τ ) ∂τ

(46)

can therefore be solved as ∂

R(y, t ) = exp(y(e ∂τ − 1))a(τ ) =

∞ X 1 Ben (y)a(n) (τ ). n ! n=0

(47)

We must underline that we can have on the l.h.s. of Eq. (46) any other operator b Oτ and its solution can always be formally written in terms of an exponential operator leading to an expansion in terms of Bell polynomials. Let us finally consider the lower factorial polynomials, which are Sheffer generated by ∞ n X t n =0

n!

γn (x) = (1 + t )x

γn (x) =

(48)

Γ (x + 1) , Γ (x + 1 − n)

and with monomiality operators

b = x exp(−b M Dx ) b P = exp(b D) − 1.

(49)

This case is particularly interesting since it allows the solution of differential difference problems of the type

∂ F (x, τ ) = xF (x + 1, τ ) + [F (x + 1, τ ) − F (x, τ )] ∂τ F (x, 0) = g (x),

(50)

which can be formally reduced to the same problem written in Eq. (30), thus getting the solution in the form 1 2

F (x, τ ) = e− 2 τ eτ x exp(−Dx ) g (x) b

 X ∞ ∞ 1 Ben (τ ) X τ s (n) = exp − τ 2 hs (x), 2 n! s! n =0 s=0

(51)

hs+1 (x) = xhs (x + 1), h1 (x) = xg (x + 1) = (x)s g (x + s), (x)s = x(x + 1) · · · (x + s), and the suffix (n) denotes the n-th order derivative. References [1] See e.g. K.B. Wolf, Representations of the algebra sp(2, R). Lie methods in optics (León, 1985), in: Lecture Notes in Phys., vol. 250, Springer, Berlin, 1986, pp. 239–247. [2] G. Dattoli, Hermite–Bessel and Laguerre–Bessel functions a by product of the monomiality principle, in: D. Cocolicchio, G. Dattoli, H.M. Srivastava, (Eds.) Advanced special functions and applications (Melfi, 1999), in: Proc. Melfi Sch. Adv. Top. Math., vol. 1, Rome, 2000, pp. 147–164. [3] G. Dattoli, B. Germano, M.R. Martinelli, P.E. Ricci, Monomiality, orthogonal and pseudo-orthogonal polynomials, Int. Math. Forum 1 (2006) 603–616. [4] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, in: Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1984, Halsted Press (John Wiley & Sons, Inc.), New York. [5] G. Dattoli, H.M. Srivastava, K. Zhukovsky, Operational methods and differential equations, with application to initial value problems, Appl. Math. Comput. 184 (2007) 979–1001. [6] P. Blasiak, G. Dattoli, A. Horzela, K.A. Penson, Representations of monomiality principle with Sheffer-type polynomials and boson normal ordering, Phys. Lett. A 352 (1–2) (2006) 7–12.