fulltext

NOTE BREVI

Quasi-monomials and isospectral problems

G. DATTOLI, A. TORRE and G. MAZZACURATI(*)

ENEA, Dipartimento Innovazione, Divisione Fisica Applicata, C. R. Frascati C.P. 65, 00044 Frascati, Roma, Italy

(ricevuto il 26 Giugno 1996; approvato il 30 Ottobre 1996)

Summary. — We provide examples of quasi-monomials for differential and finite-difference operators. We show that classes of isospectral problems can be generated and that new types of functions, constructed in terms of quasi-monomials, can be defined.

PACS 02.30.Gp – Special functions.

PACS 02.30.Hq – Ordinary differential equations.

The Hermite polynomials Hen(x), usually introduced through the generating

function [1] ext 21₂t2 4

!

!

y

z

and satisfy the ricurrence relations d

dxHen(x) 4nHen 21(x) ,

g

dx

h

According to eq. (3), we can view the Hermite polynomials as “quasi-monomials” under the action of the operators d

dx , x 2 d

dx. In fact the derivative acts formally on Hen(x)

as on the monomials xn

(n D0) and the operator x2 _dxd plays the role of multiplication operator.

(*) ENEA student.

Since the commutator of p × 4 d dx , m× 4x2 d dx (4)

is equal to one, we can conclude that p× and m× form a representation of the Heisenberg algebra. One possible interpretation of [2] is that of identifying p× with the momentum and m× with the conjugate coordinate.

We can take advantage from the previously established correspondence, namely d

dx K p× , x× Km× , x

n

K Hen(x)

(5)

to reconsider well-known eigenvalue problems, e.g.

L

k

dx , x

l

and derive classes of isospectral operators, namely L[p× , m×] f(m×) 4nf(m×) . (7)

If f admits a series expansion we can write [2] f(m×) 4

!

g

h

!

u

v

If we consider f as a function and not as an operator, we obtain

hf(x) 4

!

ncnHen(x)

(9)

in complete agreement with the identification of Hen(x) as quasi-monomials.

Accordingly, we can redefine a known function in terms of quasi-monomials, by substituting xn_{with He}

n(x) in its Taylor expansion. This definition makes sense if the

convergence is ensured. By denoting this type of function, with the suffix h, we note that the analogue of the exponential is given by

hex4

!

and that it is an eigenfunction of p× .

It is now interesting to note that, since the Hen(x) satisfy the eigenvalue

equation

g

h

their h-analogue satisfy the equation

g

h

and are provided by

hHen(x) 4

!

y

z

By using the same principle, we can define h-polynomials starting from the well-known cases and study the corresponding isospectral problem. The h-analogue of the Laguerre polynomials [1] is indeed given by

hLn(x) 4

!

and after a little algebra one can prove that [m×p×2

1 ( 1 2 m×) p_{× 1n]}hLn(x) 40

(15)

which, in terms of differential operators writes

y

z

We have obtained, by using the substitutions summarized in eqs. (5), a new classes of polynomials which satisfy a third-order differential equation.

We can extend the method and introduce, e.g., h-Bessel functions, by using the following series definition:

hJn(x) 4

!

whose convergence can be easily established.

It can be also proved that the h-Bessel functions of first kind (1) satisfy the

(1_{) To be more precise eq. (17) identifies a first-kind cylinder h-Bessel functions. We could also}

define its modified forms as

hIn(x) 4

!

Q _He

n 12s(x)

2n 12s_{s!(n 1s)!} .

In this case the identity [1]

In(ix) 4inJn(x)

recurrence relations (18) 2 p×

hJn(x) 4 [hJn 21(x) 2hJn 11(x) ] , 2 nhJn(x) 4m× [hJn 21(x) 1hJn 11(x) ] ,

which can be handled to obtain the following fourth-order differential equation: (m×p×m×p_{× 1m}×2

2 n2)hJn(x) 40

(19)

which in terms of differential operators reads

y

z

It is worth to clarify the following points: can we interpret the h-functions in terms of known functions?

This is possible in same cases only. The h-exponential can be indeed identified as

hex4 ex 2

1 2 . (21)

Furthermore, it is also easy to show that the generating function of the h-Hermite polynomials is given by ext 2t2 4

!

which can be identified as a particular case of the Kampé de Feriet polynomials [3] provided by the following generating function:

ext 1yt2₄

!

g

h

so a straightforward interpretation ofhLn(x) polynomials is not possible and this holds

for the hJn(x) functions too (2). A more systematic analysis will be presented

elsewhere.

Before closing this note, we want to stress that the use of the concept of quasi-monomial is very useful and can be exploited in different contexts.

In the case of the finite-difference formalism, one introduces the operators [4] D×₄ e

d( d Odx)21

d , x× 4xe

d( d Odx)

(24)

(2) ThehJn(x) belong to particular classes of generalized Bessel functions which will be discussed

and the associated quasi-monomials are specified by x_d(n 11)₄

»

g

h

g

h

all the previous discussion holds in this case too. We find indeed f (x_{×) 4}

!

!

and if f is not considered as an operator we obtain that it is a quasi-monomial function. To give a further example, we note that the “discrete” version of the Laguerre polynomials is given by dLn(x) 4

!

and that it satisfied the eigenvalue equation [x×D×2

1 ( 1 2 x×) D×1 n]dLn(x) 40 .

(28)

We go further in the definition of quasi-monomial structures, and introduce forms which generalized the already discussed cases.

According to the previous discussion, the d-Hermite polynomials are defined as follows: dHen(x) 4

!

y

z

and satisfy the recurrences

D×[dHen(x) ] 4n[dHen 21(x) ] , (x× 2D×)[dHen(x) ] 4dHen 11(x) .

(30)

ThedHen(x) polynomials can be interpreted as quasi-monomials under the action of D×

and (x× 2D×). We can exploit this fact to define (d, h)–quasi-monomials and introduce, e.g., (d , h)-Hermite polynomials defined as

(d , h)Hen(x) 4

!

y

z

It is easy to check that the last polynomials satisfy the eigenvalue equation [x×D×_{2 2 D}×2

2 n](d , h)Hen(x) 40 .

(32)

In this note we have just outlined a few examples of the theory of quasi-monomial functions. A more complete treatment, including application to physical problems and to numerical analysis, will be presented in a dedicated monograph.

R E F E R E N C E S

[1] ANDREW L. C., Special Functions for Engineers and Applied Mathematicians (McMillan, New York, N.Y.) 1985.

[2] BURCHNALL J. S., Quart. J. Math., 12 (1941) 9.

[3] For an introduction to generalized Hermite polynomials, see, e.g., DATTOLIG. and TORREA.,

Theory and Applications of Generalized Bessel Functions (ARACNE, Roma) 1996.