Monomiality Principle and Eigenfunctions of Differential Operators

Abstract

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.

1. 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.

2. 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 𝑃𝑝_𝑛(𝑥)=𝑛𝑝_𝑛−1(𝑥),𝑀𝑝_𝑛(𝑥)=𝑝_𝑛+1(𝑥). (2.1)

The 𝑃 and 𝑀 operators are shown to satisfy the commutation property𝑃,𝑀=𝑃𝑀−𝑀𝑃=1, (2.2) 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 equation𝑀𝑃𝑝_𝑛(𝑥)=𝑛𝑝_𝑛(𝑥). (2.3)(ii)Assuming here and in the following 𝑝₀(𝑥)=1, then 𝑝_𝑛(𝑥) can be explicitly constructed as𝑝_𝑛(𝑥)=𝑀^𝑛(1). (2.4)(iii)The last identity implies that the exponential generating function of 𝑝_𝑛(𝑥) is given by 𝑒^𝑡𝑀(1)=^∞_𝑛=0𝑡𝑀^𝑛(1)=𝑛!^∞𝑛=0𝑡^𝑛𝑀𝑛!^𝑛(1), (2.5) and therefore𝑒^𝑡𝑀(1)=^∞_𝑛=0𝑡^𝑛𝑝𝑛!^𝑛(𝑥). (2.6)

3. Laguerre-Type Exponentials

For every positive integer 𝑛, the 𝑛𝐿-exponential function is defined by 𝑒_𝑛(𝑥)∶=^∞_𝑘=0𝑥^𝑘(𝑘!)^𝑛+1. (3.1) This function reduces to the classical exponential when 𝑛=0, so that we can put 𝑒₀(𝑥)∶=𝑒^𝑥.

Consider the operator (containing 𝑛+1 derivatives)𝐷_𝑛𝐿∶=𝐷𝑥⋯𝐷𝑥𝐷𝑥𝐷=𝑆(𝑛+1,1)𝐷+𝑆(𝑛+1,2)𝑥𝐷²+⋯+𝑆(𝑛+1,𝑛+1)𝑥^𝑛𝐷^𝑛+1, (3.2) 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_𝑛𝐿𝑒_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 forall𝑛, the 𝑛𝐿-exponential function satisfies 𝑒_𝑛(0)=1, and it is an increasing convex function whenever 𝑥≥0; furthermore, 𝑒^𝑥=𝑒₀(𝑥)>𝑒₁(𝑥)>𝑒₂(𝑥)>⋯>𝑒_𝑛(𝑥)>⋯,∀𝑥>0. (3.5)

According to [10], forall𝑠=1,2,3,…, it follows that (𝐷𝑥𝐷)^𝑠=𝐷^𝑠𝑥^𝑠𝐷^𝑠,(𝐷𝑥𝐷𝑥𝐷)^𝑠=𝐷^𝑠𝑥^𝑠𝐷^𝑠𝑥^𝑠𝐷^𝑠, (3.6) and so on for every 𝐷_𝑛𝐿(𝑛=1,2,3,…).

4. Eigenfunctions of Differential Operators

We start assuming 𝑎=1,𝑛=1, in (3.3), so that𝐷𝑥𝐷^∞_𝑘=0𝑥^𝑘(𝑘!)²=^∞_𝑘=0𝑥^𝑘(𝑘!)². (4.1) 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 𝑃𝑀𝑃^∞_𝑘=0𝑝_𝑘(𝑥)(𝑘!)²=^∞_𝑘=0𝑝_𝑘(𝑥)(𝑘!)². (4.2) Therefore, the operator 𝑃𝑀𝑃 admits the eigenfunction ∑^∞_𝑘=0𝑝_𝑘(𝑥)/(𝑘!)².

Proof. Searching for an eigenfunction of the form ∑^∞_𝑘=0𝑎_𝑘𝑝_𝑘(𝑥), normalized assuming 𝑎₀∶=1, we find that, by using properties (2.1), 𝑃𝑀𝑃^∞_𝑘=0𝑎_𝑘𝑝_𝑘(𝑥)=^∞_𝑘=0𝑎_𝑘𝑃𝑀𝑃𝑝_𝑘(𝑥)=^∞_𝑘=1𝑎_𝑘𝑘²𝑝_𝑘−1(𝑥)=^∞_𝑘=0𝑎_𝑘+1(𝑘+1)²𝑝_𝑘(𝑥) (4.3) and consequently, recalling 𝑎₀∶=1, 𝑎_𝑘+1𝑎_𝑘=1(𝑘+1)²⟺𝑎_𝑘=1(𝑘!)². (4.4)

By the same method, we find, for any integer 𝑛, the general result.

Theorem 4.2. The operator 𝑃𝑀𝑃⋯𝑀𝑃, including 𝑛+1 copies of the derivative operator 𝑃, admits the eigenfunction ∑^∞_𝑘=0𝑝_𝑘(𝑥)/(𝑘!)^𝑛+1, that is, 𝑃𝑀𝑃⋯𝑀𝑃^∞_𝑘=0𝑝_𝑘(𝑥)(𝑘!)^𝑛+1=^∞_𝑘=0𝑝_𝑘(𝑥)(𝑘!)^𝑛+1. (4.5)

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.

5. Hermite and Gould-Hopper Polynomials

5.1. Hermite Polynomials

Consider first the Hermite polynomials defined by the Rodrigues formula𝐻_𝑛(𝑥)=(−1)^𝑛𝑒^𝑥2𝑑^𝑛𝑑𝑥^𝑛𝑒^−𝑥2. (5.1) Recalling12𝑑𝐻𝑑𝑥^𝑛(𝑥)=𝑛𝐻_𝑛−1(𝑥),𝑑2𝑥−𝐻𝑑𝑥^𝑛(𝑥)=𝐻_𝑛+1(𝑥), (5.2) we have1𝑃=2𝑑,𝑑𝑑𝑥𝑀=2𝑥−,𝑑𝑥 (5.3) so that we find the operator𝑃𝑀1𝑃=2𝑑+1𝑑𝑥4𝑑2𝑥−𝑑𝑑𝑥𝑑𝑥²² (5.4) and the corresponding eigenfunction^+∞_𝑛=0𝐻_𝑛(𝑥)(𝑛!)²=^+∞_𝑛=0^[𝑛/2]_ℎ=0(−1)^ℎ(2𝑥)^𝑛−2ℎ().ℎ!𝑛−2ℎ!𝑛! (5.5)

5.2. Gould-Hopper Polynomials

They are defined by [11]𝐻_𝑛^(𝑚)𝑥,𝑦∶=𝑒^{𝑦𝜕𝑚𝑥}𝑥^𝑛=𝑛!^[𝑛/𝑚]_ℎ=0𝑦^ℎ𝑥^{𝑛−𝑚ℎ}()!.ℎ!𝑛−𝑚ℎ (5.6) We have in this case𝜕𝑃=,𝜕𝜕𝑥𝑀=𝑥+𝑚𝑦^𝑚−1𝜕𝑥^𝑚−1, (5.7) so that we find the operator𝑃𝑀𝜕𝑃=𝜕𝜕𝑥+𝑥²𝜕𝑥²𝜕+𝑚𝑦^𝑚+1𝜕𝑥^𝑚+1 (5.8) and the corresponding eigenfunction^∞_𝑛=0𝐻_𝑛^(𝑚)(𝑥)(𝑛!)²=^∞_𝑛=0^[𝑛/𝑚]_ℎ=0𝑦^ℎ𝑥^{𝑛−𝑚ℎ}().ℎ!𝑛−𝑚ℎ!𝑛! (5.9)

6. 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 rule𝐺_𝑛𝑥,𝑔=𝑒^𝑔(𝐷)𝑥^𝑛, (6.1) 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)=𝑥₁𝑡+𝑥₂𝑡²+⋯+𝑥_𝑟𝑡^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]𝐷𝐺_𝑛𝑥,𝑔=𝑛𝐺_𝑛−1𝑥,𝑔,𝑥+𝑔^′(𝐷)𝐺_𝑛𝑥,𝑔=𝐺_𝑛+1𝑥,𝑔. (6.2)

Therefore, they belong to the Appell class, and𝑃=𝐷_𝑥=𝐷,𝑀=𝑥+𝑔^′𝐷_𝑥=𝑥+𝑔^′(𝐷). (6.3)

As a consequence,𝑃𝑀𝑃=𝐷𝑥+𝑔^′(𝐷)𝐷=𝐷+𝑥𝐷²+𝑔^′(𝐷)𝐷²=𝐷+𝑥+𝑔^′(𝐷)𝐷², (6.4) and the corresponding eigenfunction will be_∞𝑛=0𝐺_𝑛𝑥,𝑔(𝑛!)². (6.5)

7. Bessel Functions of the First Kind

The Bessel function of the first kind,𝐽_𝑘(𝑧)=^∞_ℎ=0(−1)^ℎ(𝑧/2)^2ℎ+𝑘()!,ℎ!ℎ+𝑘 (7.1) satisfies the recurrence relations2𝑘𝑧𝐽_𝑘(𝑧)=𝐽_𝑘−1(𝑧)+𝐽_𝑘+1(𝑧),2𝑑𝐽𝑑𝑧^𝑘(𝑧)=𝐽_𝑘−1(𝑧)−𝐽_𝑘+1(𝑧). (7.2) Adding and subtracting these equalities, we get𝑑+𝑘𝑑𝑧𝑧𝐽_𝑘(𝑧)=𝐽_𝑘−1(𝑧),−𝑑+𝑘𝑑𝑧𝑧𝐽_𝑘(𝑧)=𝐽_𝑘+1(𝑧), (7.3) and, therefore, we have𝑑𝑃=𝑘+𝑘𝑑𝑧𝑧²,𝑑𝑀=−+𝑘𝑑𝑧𝑧, (7.4) so that we find the operator𝑃𝑀𝑃=−𝑘²𝑑³𝑑𝑧³+𝑘³(𝑘−1)𝑧²𝑑+𝑘𝑑𝑧³(𝑘−2)(𝑘+1)𝑧³, (7.5) and the corresponding eigenfunction^+∞_𝑘=0𝐽_𝑘(𝑧)(𝑘!)²=^+∞_𝑘=0^∞_ℎ=0(−1)^ℎ(𝑧/2)^2ℎ+𝑘ℎ!(ℎ+𝑘)!(𝑘!)². (7.6)

Note that the negative integer values of the index 𝑘 do not contribute to the series.

8. 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 𝐸₋𝑑∶=+1𝑑𝑧𝑧𝑘 (8.2) (note that 𝑧 and 𝑘 do not commute).

Applying 𝐸₋ to 𝐽_𝑛 yields 𝐸₋𝐽_𝑛(𝑧)=𝑑+1𝑑𝑧𝑧𝑘𝐽_𝑛(𝑧)=𝑑+𝑛𝑑𝑧𝑧𝐽_𝑛(𝑧)=𝐽_𝑛−1(𝑧). (8.3) Furthermore, by iteration, 𝐸₋²𝐽_𝑛(𝑧)=𝐸₋𝐸₋𝐽_𝑛(𝑧)=𝑑+𝑑𝑧𝑛−1𝑧𝑑+𝑛𝑑𝑧𝑧𝐽_𝑛(𝑧)=𝐽_𝑛−2(𝑧), (8.4) and in general 𝐸₋^𝑚𝐽_𝑛(𝑧)=𝑑+()𝑑𝑧𝑛−𝑚−1𝑧⋯𝑑+𝑛𝑑𝑧𝑧𝐽_𝑛(𝑧)=𝐽_𝑛−𝑚(𝑧). (8.5)

The derivative operator 𝑃 for the Bessel functions can be written, in terms of the number operator, as follows: 𝑃=𝑘+1𝐸₋. (8.6) In fact, by using the above rules, we have 𝑃𝐽_𝑛(𝑧)=𝑘+1𝐸₋𝐽_𝑛(𝑧)=𝑘+1𝐽_𝑛−1(𝑧)=𝑛𝐽_𝑛−1(𝑧)=𝐽_𝑛−1+𝑧2𝐽_𝑛(𝑧)+𝐽_(𝑛−2)(𝑧). (8.7) 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 𝑀∶=𝐸₊𝑑∶=−+1𝑑𝑧𝑧𝑘. (8.8) The action of the Laguerre derivative on Bessel functions becomes 𝑘+1𝐸₋𝐸₊𝑘+1𝐸₋𝐽_𝑛(𝑧)𝐽𝑛!=𝑛^𝑛−1(𝑧)()!.𝑛−1 (8.9)

The Bessel equation follows by using the factorization method: 𝐸₊𝐸₋𝐽_𝑛(𝑧)=−𝑑+𝑑𝑧𝑛−1𝑧𝑑+𝑛𝑑𝑧𝑧𝐽_𝑛(𝑧)=𝐽_𝑛(𝑧). (8.10) Note that the derivative operator can be iterated without problem, since 𝑃²𝐽_𝑛(𝑧)=𝑘+1𝐸₋𝑘+1𝐸₋𝐽_𝑛(𝑧)=𝑛(𝑛−1)𝐽_𝑛−2(𝑧), (8.11) and in general 𝑃^𝑚𝐽_𝑛(𝑧)=𝑛!()!𝐽𝑛−𝑚^𝑛−𝑚(𝑧). (8.12) 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 𝐹(𝑧)∶=^∞_𝑠=0𝜆^𝑠𝐽_𝑠(𝑧)𝑠! (8.14) is an eigenfunction of the operator 𝑃. Therefore, the operator 𝑃𝑀𝑃 admits the eigenfunction ^∞_𝑘=0𝐽_𝑘(𝑧)[𝑘!]², (8.15) 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.

9. Parabolic Cylinder Functions

We deal with the functions𝐷_𝑛(𝑧)∶=2^−(𝑛/2)𝑒^−(𝑧2/4)𝐻_𝑛𝑧√2, (9.1) where 𝐻_𝑛 denotes the ordinary Hermite polynomials (5.1). Taking into account the recurrence relations𝑧𝐷_𝑛(𝑧)=𝑛𝐷_𝑛−1(𝑧)+𝐷_𝑛+1(𝑧),𝑑𝐷𝑑𝑧^𝑛(𝑧)+𝑧2𝐷_𝑛(𝑧)=𝑛𝐷_𝑛−1(𝑧), (9.2) we have in this case [15]𝑧𝑃=2+𝑑,𝑧𝑑𝑧𝑀=2−𝑑.𝑑𝑧 (9.3) Consequently, we find the operator𝑃𝑀𝑑𝑃=−³𝑑𝑧³+𝑧²+24𝑑+𝑧𝑑𝑧³+2𝑧8, (9.4) and the corresponding eigenfunction^+∞_𝑛=0𝐷_𝑛(𝑧)(𝑛!)²=𝑒^{−𝑧2/4+∞}_𝑛=0𝐻𝑒_𝑛(𝑧)(𝑛!)². (9.5)

10. Bessel-Clifford Functions of the First Kind

They are defined by𝐶_𝑛(𝑧)1∶=Γ(𝑛+1)⁰𝐹₁(−,𝑛+1;𝑧) (10.1) and are a particular case of Wright functions [2, 16], connected with the Bessel functions of first kind by 𝐽_𝑛(𝑧)=𝑧2^𝑛𝐶_𝑛−𝑧²4. (10.2) The relevant generating function is given by 𝑒^{𝑡+(𝑧/𝑡)}=^+∞_𝑛=−∞𝐶_𝑛(𝑧)𝑡^𝑛. (10.3) From the recurrence relations𝑑𝐶𝑑𝑧^𝑛(𝑧)=𝐶_𝑛+1(𝑧),𝑧𝐶_𝑛+2(𝑧)+(𝑛+1)𝐶_𝑛+1(𝑧)=𝐶_𝑛(𝑧), (10.4) we have in this case𝑑𝑃=𝑛𝑧𝑑𝑧+𝑛²,𝑑𝑀=.𝑑𝑧 (10.5) Therefore, we find the operator𝑃𝑀𝑃=𝑛²𝑧²𝑑³𝑑𝑧³+𝑛²()𝑧𝑑𝑛+2²𝑑𝑧²+𝑛³𝑑,𝑑𝑧 (10.6) and the corresponding eigenfunction^+∞_𝑘=0𝐶_𝑘(𝑧)(𝑘!)²=^+∞_𝑘=0^+∞_ℎ=0𝑧^ℎℎ!(ℎ+𝑘)!(𝑘!)². (10.7)

11. Modified Laguerre Polynomials

They are defined by𝑓^𝛼_𝑘(𝑥)=^𝑘_ℎ=0Γ(𝑘−ℎ+𝛼)()(𝛼)𝑥ℎ!𝑘−ℎ!Γ^ℎ, (11.1) and are related (see [16]) to the classical Laguerre polynomials 𝐿_𝑛^(𝛼)(𝑥) and the Poisson-Charlier polynomials 𝑐_𝑛(𝑥;𝛼) by 𝑓_𝑘^−𝛼(𝑥)=(−1)^𝑘𝐿_𝑘^{(𝛼−𝑘)}(𝑥)=𝑥^𝑘𝑐𝑘!^𝑘(𝑥;𝛼). (11.2) We have in this case𝑑𝑃=𝑘𝑑𝑥 (11.3) coming directly from the differentiation of (11.1). Using the recurrence relation𝑓^𝛼_𝑘+1(𝑥)=1[(𝑓𝑘+1𝑘+𝑥+𝛼)]^𝛼_𝑘(𝑥)−𝑥𝑓^𝛼_𝑘−1(𝑥), (11.4) we easily obtain the operator1𝑀=[(𝑑𝑘+1𝑘+𝑥+𝛼)]−𝑥.𝑑𝑥 (11.5) Consequently, we find𝑃𝑀𝑃=𝑘²1𝑑𝑘+1+𝑑𝑥𝑥+𝛼−1𝑑𝑘+1𝑑𝑥²²𝑑−𝑥³𝑑𝑥³, (11.6) and the corresponding eigenfunction_+∞𝑘=0𝑓^𝛼_𝑘(𝑥)(𝑘!)²=^+∞_𝑘=0^𝑘_ℎ=0Γ(𝑘−ℎ+𝛼)ℎ!(𝑘−ℎ)!Γ(𝛼)(𝑘!)²𝑥^ℎ. (11.7)

12. Confluent Hypergeometric Polynomials

The Confluent Hypergeometric function Φ(𝑎,𝑐;𝑧):=₁𝐹₁(𝑎,𝑐;𝑧) reduces to a polynomial when 𝑎=−𝑚 (where 𝑚 denotes an integer number).

Recalling the recurrences [15]𝑑𝑎+𝑧Φ()(),𝑑𝑧𝑎,𝑐;𝑧=𝑎Φ𝑎−1,𝑐;𝑧𝑐−𝑎−𝑧+𝑧𝑐−𝑎𝑑𝑐−𝑎Φ()(),𝑑𝑧𝑎,𝑐;𝑧=Φ𝑎+1,𝑐;𝑧 (12.1) we have in this case𝑑𝑃=𝑎+𝑧,𝑑𝑧𝑀=𝑐−𝑎−𝑧+𝑧𝑐−𝑎𝑑𝑐−𝑎,𝑑𝑧 (12.2) so that putting again 𝑎=−𝑚 we find the operator𝑃𝑀1𝑃=𝑧𝑐−𝑎³𝑑³𝑑𝑧³+𝑧²()𝑑𝑐+3−𝑧²𝑑𝑧²)]𝑑+𝑧[(𝑐−𝑎−𝑧+1)(𝑎+1−𝑧)𝑑𝑧+𝑎[(𝑐−𝑎−𝑧𝑎−𝑧] (12.3) and the corresponding eigenfunction^+∞_𝑚=0Φ(−𝑚,𝑐;𝑧)(𝑚!)². (12.4)

13. Hypergeometric Polynomials

A similar result holds for the Hypergeometric polynomials 𝐹(𝑎,𝑏,𝑐;𝑧):=₂𝐹₁(𝑎,𝑏,𝑐;𝑧) when 𝑎=−𝑚 (integer number).

Recalling the recurrences [15]𝑑𝑎+𝑧𝐹()(),()𝑑𝑧𝑎,𝑏,𝑐;𝑧=𝑎𝐹𝑎+1,𝑏,𝑐;𝑧𝑎−𝑐+𝑏𝑧−𝑧()𝑎−𝑐1−𝑧𝑑𝑎−𝑐𝐹()(),𝑑𝑧𝑎,𝑏,𝑐;𝑧=𝐹𝑎−1,𝑏,𝑐;𝑧 (13.1) we have in this case 𝑧𝑀=1+𝑎𝑑,()𝑑𝑧𝑃=𝑎𝑧1−𝑧𝑑𝑐−𝑎−𝑑𝑧𝑏𝑎𝑧𝑐−𝑎+𝑎, (13.2) so that putting again 𝑎=−𝑚 we find the operator𝑃𝑀𝑎𝑃=(𝑐−𝑎)²𝑧³(1−𝑧)²𝑑³𝑑𝑧³+𝑧²((𝑑1−𝑧)[𝑐+3−𝑧𝑎+𝑏+5)]²𝑑𝑧²)(𝑑+𝑧[(1−𝑧)(𝑎+1)(1−2𝑧+𝑐−𝑎−𝑏𝑧+𝑧𝑎+𝑏−𝑐−2+2𝑧)]+[()(𝑑𝑧𝑏𝑧𝑎+1)(2𝑧−𝑐+𝑎+1+𝑎𝑐−𝑎)(𝑐−𝑎−𝑏𝑧)] (13.3) and the corresponding eigenfunction^+∞_𝑚=0𝐹(−𝑚,𝑏,𝑐;𝑧)(𝑚!)². (13.4)

14. 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 𝐷_𝐿+𝑚𝐷=𝐷𝑥𝐷+𝑚𝐷=𝐷(𝑥𝐷+𝑚) (14.1) (𝑚 real or complex constant), or more generally the operator 𝐷(𝑥𝐷+𝑚)^𝑛=^𝑛_𝑘=0𝑛𝑘𝐷_𝑘𝐿𝑚^𝑛−𝑘 (14.2) (𝑛 positive integral number), and the corresponding eigenfunctions ^∞_𝑘=0𝑥^𝑘()!,𝑘!𝑘+𝑚 (14.3) or ^∞_𝑘=0𝑥^𝑘(𝑘!)^𝑛()!.𝑘+𝑚 (14.4) The monomiality principle ensures that, by considering a quasi-monomial system {𝑝_𝑘(𝑥)} and the relevant derivative 𝑃 and multiplication 𝑀, the operators 𝑃𝑀𝑃+𝑚𝑃, or more generally 𝑃(𝑀𝑃+𝑚𝐼)̂^𝑛, admit the eigenfunctions _∞𝑘=0𝑝_𝑘(𝑥)𝑘!(𝑘+𝑚)! (14.5) or _∞𝑘=0𝑝_𝑘(𝑥)(𝑘!)^𝑛()!.𝑘+𝑚 (14.6) The explicit expression of these operators in the case of several quasi-monomial systems will be considered in a forthcoming paper.

Acknowledgments

The authors are grateful to Professor Dr. G. Dattoli for providing us the proof contained in Section 8. The support by the research unit “Matemática e Aplicações” from the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), is gratefully acknowledged.