Monomiality Principle and Eigenfunctions of Differential Operators

We apply the so-called monomiality principle in order to construct
eigenfunctions for a wide set of ordinary differential operators, relevant
to special functions and polynomials, including Bessel functions and
generalized Gould-Hopper polynomials.


Introduction
In many paper the so-called monomiality principle, introduced by Dattoli et al. 1 , was used in order to study in a standard way the most important properties of special polynomials and functions 2 .
In this paper, we show that the abstract framework of monomiality can be used even to find in a constructive way the eigenfunctions of a wide set of linear differential operators connected with the Laguerre-type exponentials introduced in 3 .In this paper, we limit ourselves to consider the first Laguerre derivative D L : DxD, so that we substitute the derivative D and multiplication operator x. with the corresponding derivative and multiplication operators P and M, relevant to a given set of special polynomials or functions.
The same procedure could be generalized by considering for any integer n the higher-order Laguerre derivatives D nL : Dx • • • DxDxD containing n 1 ordinary derivatives , showing that this method can be used to obtain eigenfunctions for each one of the infinite many operators obtained by using the same substitutions described before.
It can be noticed that this gives a further proof of the power of the monomiality technique.

The Monomiality Principle
The idea of monomiality traces back to Steffensen 4 , who suggested the concept of poweroid, but only recently this idea was systematically used by Dattoli 2 .It was shown in 5 that all polynomial families are essentially the same, since it is possible to obtain one of them by transforming each other by means of suitable operators, called derivative and multiplication operators.However, the derivative and multiplication operators, relevant to a general polynomial set, are expressed by formal series of the derivative operator, so that it is in general impossible to obtain sufficiently simple formulas to work with.
However, for particular polynomials sets, relevant to suitable generating functions, the above-mentioned formal series reduce to finite sums, so that their main properties can be easily derived.The leading set in this field is given by the Hermite-Kampé de Fériet shortly H-KdF also called Gould-Hopper polynomials 6, 7 .
Following Dattoli 2 , we start with the following definition.
Definition 2.1.The polynomial set {p n x } n∈N is a quasimonomial set if there exist two linear operators P and M, called, respectively, derivative operator and multiplication operator, verifying ∀n ∈ N the identities

2.1
The P and M operators are shown to satisfy the commutation property and thus display a Weyl group structure.If the considered polynomial set {p n x } is a quasi-monomial set, then its properties can be easily derived from those of the P and M operators.In fact the following holds.
i If P and M have a differential realization, then the polynomial p n x satisfies the differential equation ii Assuming here and in the following p 0 x 1, then p n x can be explicitly constructed as iii The last identity implies that the exponential generating function of p n x is given by and therefore 2.6

Laguerre-Type Exponentials
For every positive integer n, the nL-exponential function is defined by e n x : This function reduces to the classical exponential when n 0, so that we can put e 0 x : e x .Consider the operator containing n 1 derivatives where S n 1, 1 , S n 1, 2 , . . ., S n 1, n 1 denote Stirling numbers of the second kind.In 3 see also 8, 9 for applications , the following theorem is proved.
Theorem 3.1.Let a be an arbitrary real or complex number.The nth Laguerre-type exponential e n ax is an eigenfunction of the operator D nL , that is, D nL e n ax ae n ax .

3.3
For n 0, we have D 0L : D, and therefore 3.3 reduces to the classical property of the exponential function De ax ae ax .

3.4
It is worth noting that for all n, the nL-exponential function satisfies e n 0 1, and it is an increasing convex function whenever x ≥ 0; furthermore, 3.5

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According to 10 , for all s 1, 2, 3, . .., it follows that and so on for every D nL n 1, 2, 3, . . . .

Eigenfunctions of Differential Operators
We start assuming a 1, n 1, in 3.3 , so that By applying the monomiality principle to 4.1 , we find the following result.
Theorem 4.1.Let {p k x } be a polynomial (or function) set, and denote by P and M the corresponding derivative and multiplication operators.Then Therefore, the operator P M P admits the eigenfunction Proof.Searching for an eigenfunction of the form ∞ k 0 a k p k x , normalized assuming a 0 : 1, we find that, by using properties 2.1 , and consequently, recalling a 0 : 1, 4.4 By the same method, we find, for any integer n, the general result.
Theorem 4.2.The operator P M P • • • M P , including n 1 copies of the derivative operator P , admits the eigenfunction ∞ k 0 p k x / k! n 1 , that is, We want to show, in the following sections, several examples of this method, deriving explicit eigenfunctions for a large set of differential operators, connected with classical polynomial or function sets.

Hermite Polynomials
Consider first the Hermite polynomials defined by the Rodrigues formula we have 3 so that we find the operator and the corresponding eigenfunction

Gould-Hopper Polynomials
They are defined by 11

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We have in this case so that we find the operator and the corresponding eigenfunction 5.9

Generalized Gould-Hopper Polynomials
In 12 a general set of polynomials, generalizing the Gould-Hopper ones is introduced.For shortness we will call them GGHP.They are defined by the operational rule where g t is an analytic function and D : D x d/dx.Of course, if g t yt m , then the Gould-Hopper polynomials are recovered.It is worth noting that if g t is a polynomial vanishing at t 0, assuming g t x 1 t x 2 t 2 • • • x r t r , the GGHP give back the many-variable one-index Hermite polynomials see, e.g., 13 .Extensions of the last ones to many indices are given in 14 .
The G n x, g satisfy 12

6.2
Therefore, they belong to the Appell class, and As a consequence, and the corresponding eigenfunction will be 6.5

Bessel Functions of the First Kind
The Bessel function of the first kind,

7.2
Adding and subtracting these equalities, we get and, therefore, we have so that we find the operator and the corresponding eigenfunction Note that the negative integer values of the index k do not contribute to the series.

A Direct Proof for Bessel Functions
Considering the case of Bessel functions J n , we could proceed as follows.
a We define the "number operator" k associated to them by putting kJ n z : nJ n z .8.1 b And we define the relevant shift-down operator by note that z and k do not commute .Applying E − to J n yields Furthermore, by iteration, and in general The derivative operator P for the Bessel functions can be written, in terms of the number operator, as follows: In fact, by using the above rules, we have Since the number operator does not commute with z and d/dz, it does not commute with P too.
c We define the shift-up operator by The action of the Laguerre derivative on Bessel functions becomes The Bessel equation follows by using the factorization method: Note that the derivative operator can be iterated without problem, since and in general Note that E − J n λz λJ n−1 λz .

8.13
By using the preceding equations, it is easy to see that the function is an eigenfunction of the operator P .Therefore, the operator P M P admits the eigenfunction as can be checked directly.
Of course it should be very complicated to write similar equations for each monomial set, but this is useless, since the abstract Theorem 4.1 guarantees the validity of our result.

Parabolic Cylinder Functions
We deal with the functions where H n denotes the ordinary Hermite polynomials 5.1 .Taking into account the recurrence relations we have in this case 15 Consequently, we find the operator and the corresponding eigenfunction He n z n! 2 .9.5

Bessel-Clifford Functions of the First Kind
They are defined by and are a particular case of Wright functions 2, 16 , connected with the Bessel functions of first kind by The relevant generating function is given by From the recurrence relations Therefore, we find the operator and the corresponding eigenfunction 10.7

Modified Laguerre Polynomials
They are defined by We have in this case coming directly from the differentiation of 11.1 .Using the recurrence relation we easily obtain the operator 11.5

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Consequently, we find and the corresponding eigenfunction

Confluent Hypergeometric Polynomials
The Confluent Hypergeometric function Φ a, c; z : 1 F 1 a, c; z reduces to a polynomial when a −m where m denotes an integer number .
Recalling the recurrences 15 so that putting again a −m we find the operator and the corresponding eigenfunction 12.4

Hypergeometric Polynomials
A similar result holds for the Hypergeometric polynomials F a, b, c; z : 2 F 1 a, b, c; z when a −m integer number .
Recalling the recurrences 15 we have in this case so that putting again a −m we find the operator and the corresponding eigenfunction 13.4

Conclusion
The above consideration shows that, even in the most simple case of the first-order Laguerre derivative D L : DxD, the use of monomiality gives us the possibility to construct explicitly eigenfunctions for a wide set of linear differential operators, by using a very simple and standard method.The extension of this method to the higher-order Lagaerre derivatives D nL : Dx • • • DxDxD containing n 1 ordinary derivatives could be obtained in a similar way; however, the manual computation becomes very hard when the order n increases.The use of symbolic computer algebra programs like Mathematica could be exploited in order to obtain the relevant formulas in an easy way.However, we think that this extension does not add further elements of novelty to the above-described methodology.
Similar results can be obtained by using the operator The explicit expression of these operators in the case of several quasi-monomial systems will be considered in a forthcoming paper.

11 . 1 and
are related see 16 to the classical Laguerre polynomials L α n x and the Poisson-Charlier polynomials c n x; α by

1 International
L mD DxD mD D xD m 14.Journal of Mathematics and Mathematical Sciences m real or complex constant , or more generally the operatorThe monomiality principle ensures that, by considering a quasi-monomial system {p k x } and the relevant derivative P and multiplication M, the operators P M P m P , or more generally P