1 Introduction

Motivated by the characterization of the classical orthogonal polynomials by means of a second-order Sturm-Liouville differential equation [1], it was presented recently in [23] a symbolic approach whose aim is to describe 2-orthogonal polynomial sequences constituted by eigenfunctions of a given third-order differential operator. In this paper, we continue that pursuit, establishing a wider point of view with respect to the problem posed. Moreover, the analysis of the associated relations fulfilled by the elements of the dual sequence allows us to prove that the 2-orthogonal polynomial sequences found are also Hahn-classical.

On this subject, we must refer to the papers [10]- [12] that deal with the generalization of the Bochner’s characterization of the classical orthogonal families, providing an algebraic perspective of this framework. In [10] it is discussed a Bochner-Krall problem as a bispectral problem indicating results for continuous and discrete d-orthogonal systems. In particular, the polynomial systems presented are built from the canonical sequence by means of appropriate pairs of associative non-commutative algebras and mappings between them. Thus, using automorphisms of non-commutative algebras, the polynomial system and the differential operator are constructed simultaneously [11]. In [12], the authors developed the research from the differential operators viewpoint, by describing linear differential operators such that their sequence of eigenpolynomials satisfies a finite recurrence relation of third order, namely using difference operators.

In this paper, as in [23], we have adopted a symbolic method that aims to identify the recurrence coefficients of 2-orthogonal polynomial sequences that are eigenfunctions of a given third-order differential operator of a generic form as described in [22]. The results presented along this paper are detailed for each operator and also, several functional identities are developed. These latest will allow us to better understand the pair of functionals that define the 2-orthogonality and the Hahn-classical character of the polynomial sequences obtained, following the characterization established in [8].

This paper has the following structure: in Sect. 2 we provide the basic concepts, the most common notations and we list the results given in [22] and [23] that are required for the sequel. Section 3 is dedicated to the description of the general family found by means of the symbolic approach, and further features of the 2-orthogonal polynomials found are depicted. In Sect. 4, we prove that the general sequence previously described fulfils Hahn’s property and in the final Sect. 5, we explore further relations fulfilled by the elements of the corresponding dual sequence. These two latest sections permit the clarification of certain aspects of the polynomial coefficients of the given third-order differential operator.

2 Notations and Basic Concepts

Let \({\mathcal {P}}\) be the vector space of polynomials with coefficients in \({\mathbb {C}}\) and let \({\mathcal {P}}^{\prime }\) be its topological dual space. We denote by \(\langle u,p\rangle \) the action of the form or linear functional \(u \in {\mathcal {P}}^{\prime }\) on \(p\in {\mathcal {P}}\). In particular, \(\langle u, x^{n}\rangle :=\left( u\right) _{n}\!, n \ge 0\), represent the moments of u. In the following, a sequence \({\{P_{n}\}}_{n \ge 0}\) such that \(\deg P_{n}= n,\; n \ge 0\), (that is, for all non-negative integer n), will be called polynomial sequence (PS). Also, a PS so that all polynomials have leading coefficient equal to one will be called monic polynomial sequence (MPS).

If \({\{P_{n}\}}_{n \ge 0}\) is a MPS, there exists a unique sequence \(\{u_n\}_{n\ge 0}\), \(u_n\in {\mathcal {P}}^{\prime }\), called the dual sequence of \(\{P_{n}\}_{n\ge 0}\), such that,

$$\begin{aligned} <u_{n},P_{m}>=\delta _{n,m}\, \ n,m\ge 0. \end{aligned}$$
(2.1)

On the other hand, given a MPS \({\{P_{n}\}}_{n \ge 0}\), the expansion of \(xP_{n+1}(x)\), defines sequences in \({\mathbb {C}}\), \({\{\beta _{n}\}}_{n \ge 0}\) and \(\{\chi _{n,\nu }\}_{0 \le \nu \le n,\; n \ge 0},\) such that

$$\begin{aligned}{} & {} P_{0}(x)=1, \;\; P_{1}(x)=x-\beta _{0}, \end{aligned}$$
(2.2)
$$\begin{aligned}{} & {} xP_{n+1}(x)= P_{n+2}(x)+ \beta _{n+1}P_{n+1}(x)+\sum _{\nu =0}^{n}\chi _{n,\nu }P_{\nu }(x). \end{aligned}$$
(2.3)

Another useful presentation is the folllowing, since this relation is the result of the Euclidean division of \(P_{n+2}(x)\) by \(P_{n+1}(x)\).

$$\begin{aligned}&P_{n+2}(x) = (x-\beta _{n+1})P_{n+1}(x)-\sum _{\nu =0}^{n}\chi _{n,\nu }P_{\nu }(x)\, ,\\&P_{0}(x)=1, \;\; P_{1}(x)=x-\beta _{0}\,. \end{aligned}$$

The constant sequences \({\{\beta _{n}\}}_{n \ge 0}\) and \(\{\chi _{n,\nu }\}_{0 \le \nu \le n,\; n \ge 0}\) are often called the structure coefficients (SC) of the MPS \({\{P_{n}\}}_{n \ge 0}\) [15].

When the structure coefficients fulfil \(\chi _{n,\nu }=0\;,\; 0 \le \nu \le n-1\), \(\chi _{n,n} \ne 0\), the identities (2.2)-(2.3) refer to the well known second order recurrence associated to an orthogonal MPS. More generally, identity (2.3) may furnish a recurrence relation for a higher order corresponding to the following notion of orthogonality with respect to d given functionals.

Definition 1

[8, 17] Given \(\Gamma ^{1}, \Gamma ^{2},\ldots , \Gamma ^{d} \in {\mathcal {P}}^{\prime }\), \(d \ge 1\), the polynomial sequence \({\{P_{n}\}}_{n \ge 0}\) is called d-orthogonal polynomial sequence (d-OPS) with respect to \(\Gamma =(\Gamma ^{1},\ldots , \Gamma ^{d} )\) if it fulfils

$$\begin{aligned} \langle \Gamma ^{\alpha }, P_{m}P_{n} \rangle =0, \;\; n \ge md +\alpha ,\,\; m \ge 0, \end{aligned}$$
(2.4)
$$\begin{aligned} \langle \Gamma ^{\alpha }, P_{m}P_{md+\alpha -1} \rangle \ne 0, \;\; m \ge 0, \end{aligned}$$
(2.5)

for each integer \(\alpha =1,\ldots , d\).

Lemma 1

[16] For each \(u \in {\mathcal {P}}^{\prime }\) and each \(m \ge 1\), the two following statements are equivalent.

  1. a)

    \(\langle u, P_{m-1} \rangle \ne 0,\,\, \langle u, P_{n} \rangle =0,\, n \ge m\).

  2. b)

    \(\exists \lambda _{\nu } \in {\mathbb {C}},\,\, 0 \le \nu \le m-1,\,\, \lambda _{m-1}\ne 0\) such that \(u=\sum _{\nu =0}^{m-1}\lambda _{\nu } u_{\nu }\).

The conditions (2.4) are called the d-orthogonality conditions and the conditions (2.5) are called the regularity conditions. In this case, the functional \(\Gamma \), of dimension d, is said regular.

The d-dimensional functional \(\Gamma \) is not unique. Nevertheless, from Lemma 1, we have:

$$\begin{aligned} \Gamma ^{\alpha }=\sum _{\nu =0}^{\alpha -1}\lambda _{\nu }^{\alpha } u_{\nu }, \;\; \lambda _{\alpha -1}^{\alpha } \ne 0, \; 1\le \alpha \le d. \end{aligned}$$

Therefore, since \(U=(u_{0},\ldots , u_{d-1} )\) is unique, we use to consider the canonical functional of dimension d, \(U=(u_{0},\ldots , u_{d-1} )\), saying that \({\{P_{n}\}}_{n \ge 0}\) is d-orthogonal (for any positive integer d) with respect to \(U=(u_{0},\ldots , u_{d-1})\) if

$$\begin{aligned}&\langle u_{\nu }, P_{m}P_{n} \rangle =0, \;\; n \ge md +\nu +1,\,\; m \ge 0\,,\end{aligned}$$
(2.6)
$$\begin{aligned}&\langle u_{\nu }, P_{m}P_{md+\nu } \rangle \ne 0, \;\; m \ge 0, \end{aligned}$$
(2.7)

for each integer \(\nu =0,1,\ldots , d-1\).

Theorem 1

[17] Let \({\{P_{n}\}}_{n \ge 0}\) be a MPS. The following assertions are equivalent:

  1. a)

    \({\{P_{n}\}}_{n \ge 0}\) is d-orthogonal with respect to \(U=(u_{0},\ldots , u_{d-1})\).

  2. b)

    \({\{P_{n}\}}_{n \ge 0}\) satisfies a \((d+1)\)-order recurrence relation (\(d \ge 1\)):

    $$\begin{aligned} P_{m+d+1}(x)=(x-\beta _{m+d})P_{m+d}(x)-\sum _{\nu =0}^{d-1}\gamma _{m+d-\nu }^{d-1-\nu }P_{m+d-1-\nu }(x), \,\, m \ge 0, \end{aligned}$$

    with initial conditions

    $$\begin{aligned} P_{0}(x)=1,\,\, P_{1}(x)=x-\beta _{0}\;\, \text {and if }d \ge 2 \,\, \text {then}\, \\ P_{n}(x)=(x-\beta _{n-1})P_{n-1}(x)-\sum _{\nu =0}^{n-2}\gamma _{n-1-\nu }^{d-1-\nu }P_{n-2-\nu }(x), \,\, 2 \le n \le d, \end{aligned}$$

    and regularity conditions: \(\gamma _{m+1}^{0} \ne 0,\, \; m \ge 0.\)

Finally, given \(\varpi \in {\mathcal {P}}\) and \(u \in {\mathcal {P}}'\), the form \(\varpi u\), called the left-multiplication of u by the polynomial \(\varpi \), is defined by

$$\begin{aligned} \langle \varpi u,p \rangle = \langle u,\varpi p \rangle ,\,\,\, \forall p \in {\mathcal {P}}, \end{aligned}$$
(2.8)

and the transpose of the derivative operator on \({\mathcal {P}}\) defined by \(p \rightarrow (Dp)(x)=p^{\prime }(x),\) is the following (cf. [15]):

$$\begin{aligned} u \rightarrow Du:\;\;\langle Du,p \rangle =-\langle u,p^{\prime } \rangle ,\,\,\, \forall p \in {\mathcal {P}},\end{aligned}$$
(2.9)

so that we can easily establish that

$$\begin{aligned}&D(pu)=p^{\prime }u+pD(u). \end{aligned}$$
(2.10)

In this paper, we will focus on 2-orthogonal MPS, thus fulfilling the third order recurrence relation:

$$\begin{aligned}&P_{n+3}(x)=(x-\beta _{n+2})P_{n+2}(x)-\alpha _{n+2}P_{n+1}(x)-\gamma _{n+1}P_{n}(x),\end{aligned}$$
(2.11)
$$\begin{aligned}&P_{0}(x)=1,\,\, P_{1}(x)=x-\beta _{0},\,\, P_{2}(x)=(x-\beta _{1})P_{1}(x)-\alpha _{1}\, ,\; \,n \ge 0. \end{aligned}$$
(2.12)

In a final section of the paper, it is required the use of a Lemma of page 297 of [19] that has the following content.

Lemma 2

[19] Let M and N be two polynomials such that \(Mu_{0}=Nu_{1}\). If the vector functional \(U=(u_{0}, u_{1})^{T}\) is regular, then necessarily \(M=0\) and \(N=0\).

2.1 Differential Operators on \({\mathcal {P}}\) and Technical Identities

In this subsection, we list the main results indicated in [22] and [23] that will be needed throughout this text.

Given a sequence of polynomials \(\{a_{\nu }(x)\}_{ \nu \ge 0}\), let us consider the following linear mapping \(J: {\mathcal {P}} \rightarrow {\mathcal {P}}\) (cf. [20, 26]).

$$\begin{aligned} J= \sum _{\nu \ge 0} \frac{a_{\nu }(x)}{\nu !} D^{\nu },\quad \deg a_{\nu } \le \nu ,\quad \nu \ge 0. \end{aligned}$$
(2.13)

Expanding \(a_{\nu }(x)\) as follows:

$$\begin{aligned} a_{\nu }(x)=\sum _{i=0}^{\nu } a_{i}^{[\nu ]}x^{i}, \end{aligned}$$

and recalling that \( D^{\nu }\left( \xi ^n \right) (x)= \frac{n!}{(n-\nu )!} x^{n-\nu } \), we get the next identities about J:

$$\begin{aligned}{} & {} J\left( \xi ^n \right) (x) = \sum _{\nu = 0}^{n} a_{\nu } (x) \left( {\begin{array}{c}n\\ \nu \end{array}}\right) x^{n-\nu }\,, \end{aligned}$$
(2.14)
$$\begin{aligned}{} & {} J\left( \xi ^n \right) (x) = \sum _{\tau = 0}^{n} \left( \sum _{\nu =0}^{\tau }\left( {\begin{array}{c}n\\ n-\nu \end{array}}\right) a_{\tau -\nu }^{[n-\nu ]} \right) x^{\tau },\quad n \ge 0. \end{aligned}$$
(2.15)

In particular, a linear mapping J is an isomorphism if and only if

$$\begin{aligned} \deg \left( J\left( \xi ^n \right) (x) \right) = n, \;\; n \ge 0, \; \; \text {and}\;\; J\left( 1 \right) (x) \ne 0. \end{aligned}$$
(2.16)

The next result indicates that any operator that does not increase the degree admits an expansion as (2.13) for certain polynomial coefficients.

Lemma 3

[22] For any linear mapping J, not increasing the degree, there exists a unique sequence of polynomials \(\{a_{n}\}_{n \ge 0}\), with \(\deg a_{n} \le n\), so that J is read as in (2.13). Further, the linear mapping J is an isomorphism of \({\mathcal {P}}\) if and only if

$$\begin{aligned} \sum _{\mu =0}^{n}\left( {\begin{array}{c}n\\ \mu \end{array}}\right) a_{\mu }^{[\mu ]} \ne 0, \quad n \ge 0. \end{aligned}$$
(2.17)

The polynomial \(J\left( fg\right) \) is given by a Leibniz-type development [22] as mentioned in the next Lemma.

Lemma 4

[22] For any \(f,g \in {\mathcal {P}}\), we have:

$$\begin{aligned}&J\left( f(x)g(x) \right) (x)= \sum _{n \ge 0 } J^{(n)}\left( f\right) (x) \frac{g^{(n)}(x)}{n!} = \sum _{n \ge 0 } J^{(n)}\left( g\right) (x) \frac{f^{(n)}(x)}{n!}, \end{aligned}$$
(2.18)

where the operator \(J^{(m)}\,, \; m \ge 0\), on \({\mathcal {P}}\) is defined by

$$\begin{aligned} J^{(m)}= \displaystyle \sum _{n \ge 0 } \frac{a_{n+m}(x)}{n!}D^{n}, \end{aligned}$$
(2.19)

whose transposed operator is given by [22]

$$\begin{aligned} J^{(m)}(u) = \sum _{n \ge 0 } \frac{(-1)^n}{n!}D^n\left( a_{n+m} u \right) , \quad m \ge 0. \end{aligned}$$
(2.20)

Let us suppose that J is an operator expressed as in (2.13), and acting as the derivative of order k, for some non-negative integer k, that is, it fulfils the following conditions.

$$\begin{aligned}&J\left( \xi ^{k}\right) (x) = a_{0}^{[k]} \ne 0 \;\; \text {and} \; \;\deg \left( J\left( \xi ^{n+k}\right) (x) \right) = n,\quad n \ge 0; \end{aligned}$$
(2.21)
$$\begin{aligned}&J\left( \xi ^{i}\right) (x) =0,\quad 0 \le i \le k-1,\; \text {if} \; k\ge 1. \end{aligned}$$
(2.22)

Lemma 5

[22] An operator J fulfils (2.21)-(2.22) if and only if the next conditions hold.

  1. a)

    \(a_{0}(x)=\cdots =a_{k-1}(x)=0\), if \(k \ge 1\);

  2. b)

    \(\deg \left( a_{\nu }(x) \right) \le \nu -k\), \(\nu \ge k\);

  3. c)
    $$\begin{aligned} \lambda _{n+k}^{[k]}:= \sum _{\nu = 0}^{n} \left( {\begin{array}{c}n+k\\ n+k-\nu \end{array}}\right) a_{n-\nu }^{[n+k-\nu ]} \ne 0, \; n \ge 0. \end{aligned}$$
    (2.23)

Remark 1

Note that in (2.23) we find \(\lambda _{k}^{[k]}=a_{0}^{[k]} \).

If \(k=0\), then it is assumed that \(\lambda _{n}^{[0]} \ne 0,\; n \ge 0\), matching (2.17), so that J is an isomorphism.

If \(k=1\), then J imitates the usual derivative and is commonly called a lowering operator (e.g. [14, 21]).

Applying Lemma 4 to different pairs of polynomials, we obtain immediately the next identities.

$$\begin{aligned}&J\left( xp(x)\right) = xJ\left( p(x)\right) +J^{(1)}\left( p(x) \right) \end{aligned}$$
(2.24)
$$\begin{aligned}&J\left( x^2p(x)\right) = x^2J\left( p(x)\right) +2xJ^{(1)}\left( p(x) \right) +J^{(2)}\left( p(x) \right) \end{aligned}$$
(2.25)
$$\begin{aligned}&J\left( x^3p(x)\right) =x^3J\left( p(x)\right) +3x^2J^{(1)}\left( p(x) \right) +3x J^{(2)}\left( p(x) \right) + J^{(3)}\left( p(x) \right) \end{aligned}$$
(2.26)

Proposition 2

[23] Given an operator J defined by (2.13), and taking into account the definition of the operator \(J^{(m)}\), \( m \ge 0\), on \({\mathcal {P}}\):

$$\begin{aligned} J^{(m)}= \displaystyle \sum _{n \ge 0 } \frac{a_{n+m}(x)}{n!}D^{n}, \end{aligned}$$

the following identities hold.

$$\begin{aligned}&J^{(i)}\left( x p(x) \right) = J^{(i+1)}\left( p(x) \right) +x J^{(i)}\left( p(x) \right) \, , \, \; i=0,1,2,\ldots . \end{aligned}$$
(2.27)

Lemma 6

[23] Given an operator J defined by (2.13), and supposing that \(\{ P_{n}(x) \}_{n \ge 0}\) is a 2-orthogonal MPS, the following identities hold.

$$\begin{aligned} J^{(2)}\left( P_{n+2}(x) \right)&= J^{(3)}\left( P_{n+1}(x) \right) +x J^{(2)}\left( P_{n+1}(x) \right) \\ \nonumber&-\beta _{n+1}J^{(2)}\left( P_{n+1}(x) \right) -\alpha _{n+1}J^{(2)}\left( P_{n}(x) \right) -\gamma _{n}J^{(2)}\left( P_{n-1}(x) \right) \, ;\end{aligned}$$
(2.28)
$$\begin{aligned} J^{(1)}\left( P_{n+2}(x) \right)&= J^{(2)}\left( P_{n+1}(x) \right) +x J^{(1)}\left( P_{n+1}(x) \right) \\ \nonumber&-\beta _{n+1}J^{(1)}\left( P_{n+1}(x) \right) -\alpha _{n+1}J^{(1)}\left( P_{n}(x) \right) -\gamma _{n}J^{(1)}\left( P_{n-1}(x) \right) \, ;\end{aligned}$$
(2.29)
$$\begin{aligned} J\left( P_{n+3}(x) \right)&= J^{(1)}\left( P_{n+2}(x) \right) +x J\left( P_{n+2}(x) \right) -\beta _{n+2}J\left( P_{n+2}(x) \right) \\ \nonumber&-\alpha _{n+2}J\left( P_{n+1}(x) \right) -\gamma _{n+1}J\left( P_{n}(x) \right) \, . \end{aligned}$$
(2.30)

2.2 An Isomorphism Applied to a 2-Orthogonal Polynomial Sequence

We begin by describing the assumptions taken in the results reviewed in this subsection. We consider that J is an isomorphism and \(a_{\nu }(x)=0,\; \nu \ge 4\):

$$\begin{aligned}&J = a_{0}(x)I + a_{1}(x)D+\frac{a_{2}(x)}{2!}D^{2}+\frac{a_{3}(x)}{3!}D^{3}\, ,\, \text {where}\\ \nonumber&a_{0}(x)=a_{0}^{[0]}\; , \; a_{1}(x)=a_{0}^{[1]}+a_{1}^{[1]}x \; , \; a_{2}(x)=a_{0}^{[2]}+a_{1}^{[2]}x +a_{2}^{[2]}x^2 \; , \\ \nonumber&a_{3}(x)=a_{0}^{[3]}+a_{1}^{[3]}x +a_{2}^{[3]}x^2 + a_{3}^{[3]}x^3 \;, \; \end{aligned}$$
(2.31)

and we suppose that the MPS \(\{P_{n} \}_{n \ge 0}\) is 2-orthogonal and fulfils

$$\begin{aligned} J\left( P_{n}(x) \right) = \lambda ^{[0]}_{n}P_{n}(x), \; \text {with} \; \lambda ^{[0]}_{n} \ne 0\, ,n \ge 0\, ,\end{aligned}$$
(2.32)

where

$$\begin{aligned} \lambda ^{[0]}_{n} =a_{0}^{[0]}+\left( {\begin{array}{c}n\\ 1\end{array}}\right) a_{1}^{[1]}+\left( {\begin{array}{c}n\\ 2\end{array}}\right) a_{2}^{[2]}+ \left( {\begin{array}{c}n\\ 3\end{array}}\right) a_{3}^{[3]}\,, \, n \ge 0 . \end{aligned}$$

In view of \(a_{\nu }(x)=0,\; \nu \ge 4\,,\) the operators on \({\mathcal {P}}\), \(J^{(1)}\), \(J^{(2)}\) and \(J^{(3)}\) have the following definitions as indicated in (2.19).

$$\begin{aligned}&J^{(1)}(p) = \left( a_{1}(x)I+a_{2}(x) D +\frac{1}{2!}a_{3}(x) D^{2} \right) (p) \end{aligned}$$
(2.33)
$$\begin{aligned}&J^{(2)}(p)=\left( a_{2}(x)I+ a_{3}(x)D \right) (p)\end{aligned}$$
(2.34)
$$\begin{aligned}&J^{(3)}(p)=a_{3}(x)p\\ \nonumber&J^{(m)}(p) = 0\, , \; m \ge 4\, , \quad p \in {\mathcal {P}}. \end{aligned}$$
(2.35)

Next, we review the main identities obtained in the symbolic implementation described in [23], considering henceforth \(P_{-i}(x)=0\), \(i=1,2,\ldots \).

First step: applying J to the third order recurrence

$$\begin{aligned} J^{(1)}\left( P_{n}(x)\right)&= \left( \lambda ^{[0]}_{n+1} - \lambda ^{[0]}_{n}\right) P_{n+1}(x) \\ \nonumber +&\alpha _{n}\left( \lambda ^{[0]}_{n-1} - \lambda ^{[0]}_{n} \right) P_{n-1}(x)+ \gamma _{n-1} \left( \lambda ^{[0]}_{n-2}-\lambda ^{[0]}_{n} \right) P_{n-2}(x)\; , n \ge 0. \end{aligned}$$
(2.36)

Second step: applying \(J^{(1)}\) to the third order recurrence

$$\begin{aligned}&J^{(2)} \left( P_{n}(x) \right) = A_{n+2} P_{n+2}(x) + B_{n+1} P_{n+1}(x) + C_{n} P_{n}(x) \\ \nonumber&+D_{n-1} P_{n-1}(x) + F_{n-2} P_{n-2}(x)+ G_{n-3} P_{n-3}(x) + H_{n-4}P_{n-4}(x) \, , \, n \ge 0\,, \end{aligned}$$
(2.37)

where

$$\begin{aligned}&A_{n} = \lambda ^{[0]}_{n} -2 \lambda ^{[0]}_{n-1}+ \lambda ^{[0]}_{n-2}\, ;\\ \nonumber&B_{n} = \left( \beta _{n-1}-\beta _{n} \right) \left( \lambda ^{[0]}_{n}-\lambda ^{[0]}_{n-1}\right) \, ;\\ \nonumber&C_{n} = 2 \alpha _{n+1} \left( \lambda ^{[0]}_{n}- \lambda ^{[0]}_{n+1} \right) + 2 \alpha _{n} \left( \lambda ^{[0]}_{n}- \lambda ^{[0]}_{n-1} \right) \, ; \\ \nonumber&D_{n}= \alpha _{n+1} \left( \beta _{n+1}-\beta _{n} \right) \left( \lambda ^{[0]}_{n}- \lambda ^{[0]}_{n+1} \right) \\ \nonumber&\quad + \gamma _{n+1}\left( \lambda ^{[0]}_{n}-2 \lambda ^{[0]}_{n+2}+ \lambda ^{[0]}_{n+1} \right) + \gamma _{n}\left( \lambda ^{[0]}_{n}-2 \lambda ^{[0]}_{n-1}+ \lambda ^{[0]}_{n+1} \right) \, ; \\ \nonumber&F_{n} = \alpha _{n+2}\alpha _{n+1}\left( \lambda ^{[0]}_{n} -2 \lambda ^{[0]}_{n+1} + \lambda ^{[0]}_{n+2} \right) + \gamma _{n+1} \left( \beta _{n+2}-\beta _{n} \right) \left( \lambda ^{[0]}_{n} - \lambda ^{[0]}_{n+2} \right) \, ; \\ \nonumber&G_{n} = \alpha _{n+3}\gamma _{n+1} \left( \lambda ^{[0]}_{n} -2 \lambda ^{[0]}_{n+2} + \lambda ^{[0]}_{n+3} \right) + \alpha _{n+1}\gamma _{n+2} \left( \lambda ^{[0]}_{n} -2 \lambda ^{[0]}_{n+1} + \lambda ^{[0]}_{n+3} \right) \, ; \\ \nonumber&H_{n} = \gamma _{n+3}\gamma _{n+1} \left( \lambda ^{[0]}_{n} -2 \lambda ^{[0]}_{n+2} + \lambda ^{[0]}_{n+4} \right) \, . \end{aligned}$$
(2.38)

Third step: applying \(J^{(2)}\) to the third order recurrence

$$\begin{aligned}&J^{(3)} \left( P_{n+2}(x) \right) = a_{3}^{[3]}P_{n+5}(x)\\ \nonumber&+\left( A_{n+4} \beta _{n+2}-A_{n+4} \beta _{n+4}-B_{n+3}+B_{n+4} \right) P_{n+4}(x) \\ \nonumber&+\left( A_{n+3} \alpha _{n+2}-A_{n+4} \alpha _{n+4}+B_{n+3} \beta _{n+2}-B_{n+3} \beta _{n+3}-C_{n+2}+C_{n+3} \right) P_{n+3}(x)\\ \nonumber&+ \left( A_{n+2} \gamma _{n+1}-A_{n+4} \gamma _{n+3}+B_{n+2} \alpha _{n+2}-B_{n+3} \alpha _{n+3}-D_{n+1}+D_{n+2} \right) P_{n+2}(x)\\ \nonumber&+ \left( B_{n+1} \gamma _{n+1}-B_{n+3} \gamma _{n+2}+C_{n+1} \alpha _{n+2}-C_{n+2} \alpha _{n+2} \right. \\ \nonumber&\left. \hspace{2.5cm} -D_{n+1} \beta _{n+1}+D_{n+1} \beta _{n+2}-F_n+F_{n+1} \right) P_{n+1}(x)\\ \nonumber&+\left( C_n \gamma _{n+1}-C_{n+2} \gamma _{n+1}-D_{n+1} \alpha _{n+1}+D_n \alpha _{n+2} \right. \\ \nonumber&\left. \hspace{2.5cm} -F_n \beta _n+F_n \beta _{n+2}-G_{n-1}+G_n \right) P_{n}(x) \\ \nonumber&+\left( -D_{n+1} \gamma _n+D_{n-1} \gamma _{n+1}-F_n \alpha _n+F_{n-1} \alpha _{n+2} \right. \\ \nonumber&\left. \hspace{2.5cm} -G_{n-1} \beta _{n-1}+G_{n-1} \beta _{n+2}-H_{n-2}+H_{n-1} \right) P_{n-1}(x) \\ \nonumber&+\left( -F_n \gamma _{n-1}+F_{n-2} \gamma _{n+1} \right. \\ \nonumber&\left. \hspace{2.5cm} -G_{n-1} \alpha _{n-1}+G_{n-2} \alpha _{n+2}-H_{n-2} \beta _{n-2}+H_{n-2} \beta _{n+2} \right) P_{n-2}(x)\\ \nonumber&+\left( -G_{n-1} \gamma _{n-2}+G_{n-3} \gamma _{n+1}-H_{n-2} \alpha _{n-2}+H_{n-3} \alpha _{n+2} \right) P_{n-3}(x)\\ \nonumber&+ \left( H_{n-4} \gamma _{n+1}-H_{n-2} \gamma _{n-3}\right) P_{n-4}(x)\; , \; n \ge 0\, , \end{aligned}$$
(2.39)

with initial conditions:

$$\begin{aligned}&J^{(3)} \left( P_{0}(x) \right) = a_{3}^{[3]}P_{3}(x)+\left( \left( \beta _0+\beta _1+\beta _2\right) a_{3}^{[3]}+a_{2}^{[3]} \right) P_{2}(x)\\&+\left( a_{3}^{[3]} \left( \alpha _1+\alpha _2+\beta _0^2+\beta _1 \beta _0+\beta _1^2\right) +\left( \beta _0+\beta _1\right) a_{2}^{[3]}+a_{1}^{[3]} \right) P_{1}(x)\\&+\left( a^{[3]}_{3} \left( \alpha _1 \left( 2 \beta _0+\beta _1\right) +\beta _0^3+\gamma _1\right) +\alpha _1 a^{[3]}_{2}+\beta _0 \left( \beta _0 a^{[3]}_{2}+a^{[3]}_{1}\right) +a^{[3]}_{0} \right) \, ; \end{aligned}$$
$$\begin{aligned}&J^{(3)} \left( P_{1}(x) \right) = a_{3}^{[3]}P_{4}(x)+\left( \left( \beta _1+\beta _2+\beta _3\right) a_{3}^{[3]}+a_{2}^{[3]} \right) P_{3}(x)\\&+\left( a^{[3]}_{3} \left( \alpha _1+\alpha _2+\alpha _3+\beta _1^2+\beta _2 \beta _1+\beta _2^2\right) +\left( \beta _1+\beta _2\right) a^{[3]}_{2}+a^{[3]}_{1}\right) P_{2}(x)\\&+ \left( a^{[3]}_{3} \left( 2 \left( \alpha _1+\alpha _2\right) \beta _1+\alpha _2 \beta _2+\beta _1^3+\gamma _1+\gamma _2\right) +\alpha _1 \beta _0 a^{[3]}_{3}+\left( \alpha _1+\alpha _2\right) a^{[3]}_{2} \right. \\ \nonumber&\hspace{2cm} \left. +\beta _1 \left( \beta _1 a^{[3]}_{2}+a^{[3]}_{1}\right) +a^{[3]}_{0} \right) P_{1}(x)\\ \nonumber&+\left( \alpha _1 \left( a^{[3]}_{3} \left( \alpha _2+\beta _0^2+\beta _1 \beta _0+\beta _1^2\right) +\left( \beta _0+\beta _1\right) a^{[3]}_{2}+a^{[3]}_{1}\right) +\alpha _1^2 a^{[3]}_{3} \right. \\ \nonumber&\hspace{2cm} \left. +\gamma _1 \left( \left( \beta _0+\beta _1+\beta _2\right) a^{[3]}_{3}+a^{[3]}_{2}\right) \right) \, . \end{aligned}$$

3 A General Family of 2-Orthogonal Polynomial Eigenfunctions

While working with the identity (2.39), taking into account that

$$\begin{aligned} J^{(3)}\left( p\right) =a_{3}(x)p= \left( a_{3}^{[3]}x^3+a_{2}^{[3]}x^2 +a_{1}^{[3]}x+a_{0}^{[3]} \right) p\,, \end{aligned}$$

we may conclude that \(\deg \left( a_{3}(x)\right) \le 2\), that is, \(a^{[3]}_{3} = 0\), which is also formally proved using the relations established in the dual space in [24].

On the other hand, either working with the initial conditions or through formal work around the elements of the dual sequence of such 2-orthogonal polynomial eigenfunctions, we are lead to infer that \(\deg \left( a_{2}(x)\right) \) cannot be equal to two. For this matter, the Corollary 1 of the final section of this text establishes that \(a^{[2]}_{2}=0\), if \(\lambda ^{[0]}_{2}\ne \lambda ^{[0]}_{1}\).

Moreover, if we suppose that \(\deg \left( a_{3}(x)\right) = 0\), we conclude that the single solution is the one described in Proposition 2 of [23]. More precisely, the search for a solution with the help of the symbolic implementation yields two situations as summarized in the next table.

\(\deg (a_{3}(x))\)

\(\deg (a_{2}(x))\)

Solutions (Y/N)

Comments

0

0

Y

\(a_{1}^{[1]} \ne 0\;\) [23]

0

1

N

[23]

When we suppose \(\deg \left( a_{3}(x)\right) = 1\) and \(\deg \left( a_{2}(x)\right) \le 1\), the simple work with the initial conditions \(J\left( P_{n}(x) \right) = \lambda _{n}^{[0]} P_{n}(x)\,, \, n =0,\ldots , 6\,,\) reduces to the additional condition \(a_{2}(x)=0\) which is already established as impossible in [23].

Therefore, in this section we present the 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) such that

$$\begin{aligned} J\left( P_{n}(x) \right) = \lambda ^{[0]}_{n}P_{n}(x), \; \text {with} \; \lambda ^{[0]}_{n} \ne 0\, ,n \ge 0\, , \end{aligned}$$

with

$$\begin{aligned}&J = a_{0}(x)I + a_{1}(x)D+\frac{a_{2}(x)}{2}D^{2}+\frac{a_{3}(x)}{3!}D^{3}\, ,\, \text {where}\\ \nonumber&a_{0}(x)=a_{0}^{[0]}\; , \; a_{1}(x)=a_{0}^{[1]}+a_{1}^{[1]}x \; , \; a_{2}(x)=a_{0}^{[2]}+a_{1}^{[2]}x \; , \\ \nonumber&a_{3}(x)=a_{0}^{[3]}+a_{1}^{[3]}x +a_{2}^{[3]}x^2 \;,\; a_{2}^{[3]} \ne 0\,, \end{aligned}$$
(3.1)

that is, we suppose \(\deg \left( a_{3}(x)\right) = 2\) and \(\deg \left( a_{2}(x)\right) \le 1\), which in view of the above comments corresponds to the remaining general solution.

Lemma 7

If a 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) fulfils

$$\begin{aligned} J\left( P_{n}(x) \right) = \lambda ^{[0]}_{n}P_{n}(x), \; \text {with} \; \lambda ^{[0]}_{n} \ne 0\, ,n \ge 0\, , \end{aligned}$$

with J defined by (3.1), then \(a_{1}^{[1]} \ne 0\).

Proof

Let us assume that \(a_{1}^{[1]}=0\). The initial identity \(J\left( P_{1}(x) \right) = \lambda ^{[0]}_{1}P_{1}(x)\) yields \(a_{0}^{[1]}=0\). The next identity \(J\left( P_{2}(x) \right) = \lambda ^{[0]}_{2}P_{2}(x)\) implies \(a_{0}^{[2]}=0\) and \(a_{1}^{[2]}=0\). By Proposition 3 of [23], we know that if \(a_{3}(x)=a_{0}^{[3]}+a_{1}^{[3]}x +a_{2}^{[3]}x^2\) and \(a_{2}(x)=0\), then \(a_{1}^{[1]}\ne 0\) which contradicts the assumption considered and ends the proof. \(\square \)

Theorem 3

Let us consider a 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) fulfilling

$$\begin{aligned}&J\left( P_{n}(x) \right) = \lambda _{n}^{[0]}P_{n}(x) \end{aligned}$$

where J is defined by (3.1).

Then the recurrence coefficients of the sequence \(\{P_{n}\}_{n \ge 0}\) are the following

$$\begin{aligned}&\beta _{n}= -\frac{a_{2}^{[3]}}{2 a_{1}^{[1]}}(n-1)n - \frac{a_{1}^{[2]}}{a_{1}^{[1]}}n- \frac{a_{0}^{[1]}}{a_{1}^{[1]}}\, , \, n \ge 0 \, ,\end{aligned}$$
(3.2)
$$\begin{aligned}&\alpha _{n} = \dfrac{n}{12(a_{1}^{[1]})^2} \left( -3 a_{1}^{[1]} \left( 2a_{0}^{[2]} + a_{1}^{[3]}(n-1)\right) \right. +\\ \nonumber&\quad \left. \left( a_{1}^{[2]} + a_{2}^{[3]}(n-1) \right) \left( 6a_{0}^{[1]} + (n-1) \left( 3a_{1}^{[2]} + a_{2}^{[3]}(n-2)\right) \right) \right) \, , \, n \ge 1 \, ,\end{aligned}$$
(3.3)
$$\begin{aligned}&\gamma _{n} =-\dfrac{1}{216 (a_{1}^{[1]})^3} n(n+1) \left( 36(a_{1}^{[1]})^2 a_{0}^{[3]} \right. \\ \nonumber&\quad -6 a_{1}^{[1]}\left( 6 a_{0}^{[1]}a_{1}^{[3]} + 2 (n-1) a_{1}^{[2]} a_{1}^{[3]} + \left( (1 + 2 n) a_{0}^{[2]} + (n-1)^2 a_{1}^{[3]}\right) a_{2}^{[3]}\right) \\ \nonumber&\quad \left. + a_{2}^{[3]} \left( 6 a_{0}^{[1]} + 3 n a_{1}^{[2]} + (n-1) n a_{2}^{[3]}\right) \left( 6 a_{0}^{[1]} + (n-1)\left( 3 a_{1}^{[2]} + (n-2) a_{2}^{[3]}\right) \right) \right) , \end{aligned}$$
(3.4)

\(\, n \ge 1 ,\) provided that \(\gamma _{n} \ne 0\), \(\, n \ge 1 \,;\) and the polynomial \(a_{3}(x)\) has a double root. If \(a_{1}^{[2]} \ne 0\), then the root of \(a_{2}(x)\) coincides with the root of \(a_{3}(x)\), that is,

$$\begin{aligned} a_{3}(x)= a_{2}^{[3]}\left( x +\dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \right) ^2\, \, ,\;\; a_{2}(x)= a_{1}^{[2]}\left( x +\dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \right) \, . \end{aligned}$$
(3.5)

Conversely, the 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) defined by the recurrence coefficients (3.13)-(3.15) fulfils the differential equation \(J\left( P_{n}(x) \right) = \lambda _{n}^{[0]}P_{n}(x) \,,\) \( \, n \ge 0\,,\) with J defined by (3.1), where the polynomial coefficients \(a_{3}(x)\) and \(a_{2}(x)\) fulfil (3.5) if \(a_{1}^{[2]} \ne 0\).

Proof

We compute the initial data for \(\beta _{i},\) for \(i=0,1,2,3, \) \(\alpha _{i}\, \) for \(i=1,2,3,\) and \(\gamma _{i}\), for \(i=1,2\), using the equations \(J\left( P_{j}(x)\right) -\lambda _{j}^{[0]}P_{j}(x)=0\,, \, j=1,\ldots , 4\).

The fourth iteration imposes the condition:

$$\begin{aligned} a_{1}^{[1]} \left( 2 a_{1}^{[2]} a_{0}^{[3]} -a_{0}^{[2]} a_{1}^{[3]}\right) + a_{0}^{[1]} \left( -a_{1}^{[2]} a_{1}^{[3]} + 2 a_{0}^{[2]} a_{2}^{[3]} \right) = 0\;. \end{aligned}$$

If we assume \(a_{1}^{[2]}=0\) and \(a_{0}^{[2]} \ne 0\), the initial calculations yield \(a_{2}^{[3]}=0\) which contradicts the hypotheses and it is altogether an impossible situation. In addition, \(a_{1}^{[2]}=0\) and \(a_{0}^{[2]}=0\) lead us to a case already treated in [23]. Thus, considering \(a_{1}^{[2]} \ne 0\), we may assert that

$$\begin{aligned}&a_{0}^{[3]}=\dfrac{1}{2a_{1}^{[2]}} \left( a_{0}^{[2]} a_{1}^{[3]} -\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} \left( -a_{1}^{[2]} a_{1}^{[3]} + 2 a_{0}^{[2]} a_{2}^{[3]} \right) \right) \, . \end{aligned}$$
(3.6)

In the equation (2.39) we have in the left-hand a linear combination of the set of polynomials \(\{ P_{n-2}(x), P_{n-1}(x), \ldots , P_{n+4}(x)\}\) in view of the recurrence relation fulfilled by a 2-orthogonal polynomial sequence:

$$\begin{aligned} x\left( P_{n}(x) \right) = P_{n+1}(x)+\beta _{n}P_{n}(x)+\alpha _{n}P_{n-1}(x)+\gamma _{n-1}P_{n-2}(x). \end{aligned}$$

Comparing both members of (2.39) we identify several equations, in particular, three of those allow the definition of the sequences \(\left( \beta _{n} \right) _{n \ge 0}\), \(\left( \alpha _{n} \right) _{n \ge 0}\) and \(\left( \gamma _{n} \right) _{n \ge 0}\) by this order, as follows.

Firstly, identity (2.39) establishes the equation \(\beta _{n+4}-2\beta _{n+3}+\beta _{n+2}= -\dfrac{a_{2}^{[3]}}{a_{1}^{[1]}}\) with initial data \(\beta _{2}= -\dfrac{a_{0}^{[1]}+2a_{1}^{[2]}+a_{2}^{[3]}}{a_{1}^{[1]}},\) \(\beta _{3}= -\dfrac{a_{0}^{[1]}+3a_{1}^{[2]}+3a_{2}^{[3]}}{a_{1}^{[1]}},\) yielding (3.13).

Secondly, from the knowledge of \(\left( \beta _{n} \right) _{n \ge 0}\), (2.39) indicates

$$\begin{aligned} a_{1}^{[3]} + a_{2}^{[3]} \left( \beta _{n+2}+\beta _{n+3} \right) = \left( -2\alpha _{n+4}+4\alpha _{n+3}-2\alpha _{n+2} + \left( \beta _{n+2}-\beta _{n+3} \right) ^2 \right) a_{1}^{[1]} \,. \end{aligned}$$

Using the values of the initial \(\alpha \)’s we conclude that the solution of this difference equation is given by (3.14).

Finally, the equation about \(\left( \gamma _{n} \right) _{n \ge 0}\) reads

$$\begin{aligned} \gamma _{n+3}-2\gamma _{n+2}+\gamma _{n+1} = - \dfrac{1}{3a_{1}^{[1]}}\left( a_{0}^{[3]}+ \beta _{n+2}\left( a_{1}^{[3]} +a_{2}^{[3]}\beta _{n+2}\right) +a_{2}^{[3]}\left( \alpha _{n+3}+\alpha _{n+2} \right) \right) , \end{aligned}$$

yielding (3.15), in view of \(\gamma _{1}\) and \(\gamma _{2}\).

After having defined these three sets of constants as the recurrence coefficients of the sequence \(\{P_{n}\}_{ n \ge 0}\), we must assure that all the remaining equations defined by (2.39) are also fulfilled. That only happens under the following additional conditions, that also take into account (3.6):

$$\begin{aligned} a_{1}^{[3]} = \dfrac{2a_{0}^{[2]} a_{2}^{[3]}}{a_{1}^{[2]}} \,, \; a_{0}^{[3]} = \left( \dfrac{a_{0}^{[2]}}{a_{1}^{[2]}}\right) ^2 a_{2}^{[3]} \,, \end{aligned}$$

hence justifying (3.5).

Conversely, let us assume a 2-orthogonal MPS defined by the recurrence coefficients (3.13)-(3.15) and let us consider the operator (3.1)

$$\begin{aligned} J= a_{0}(x)I + a_{1}(x)D+\frac{a_{2}(x)}{2!}D^{2}+\frac{a_{3}(x)}{3!}D^{3}\,, \end{aligned}$$

such that \(a_{2}(x)\) and \(a_{3}(x)\) fulfill (3.5), considering \(\deg \left( a_{2}(x)\right) = 1\).

It is easy to confirm that the following identities are true, for the initial values of i, like \(i=0,\ldots , nmax\) with nmax equal to 4 or 5, using for that matter the definitions of the operator J, \(J^{(1)}\) and \(J^{(2)}\), as indicated in (2.33)-(2.34).

$$\begin{aligned}&J\left( P_{i}(x) \right) = \lambda ^{[0]}_{i} P_{i}(x) \; , \, i=0,\ldots 5\, ,\\&J^{(1)}\left( P_{i}(x)\right) = \left( \lambda ^{[0]}_{i+1} - \lambda ^{[0]}_{i}\right) P_{i+1}(x) \\ +\alpha _{i}&\left( \lambda ^{[0]}_{i-1} - \lambda ^{[0]}_{i} \right) P_{i-1}(x)+ \gamma _{i-1} \left( \lambda ^{[0]}_{i-2}-\lambda ^{[0]}_{i} \right) P_{i-2}(x)\; , \, i=0,\ldots 4\, ,\\&J^{(2)} \left( P_{i}(x) \right) = A_{i+2} P_{i+2}(x) + B_{i+1} P_{i+1}(x) + C_{i} P_{i}(x) \\&+D_{i-1} P_{i-1}(x) + F_{i-2} P_{i-2}(x)+ G_{i-3} P_{i-3}(x) + H_{i-4}P_{i-4}(x) \, , \, i=0,\ldots 4\,, \end{aligned}$$

where \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(F_{i}\), \(G_{i}\) and \(H_{i}\) are defined in (2.38).

As induction hypotheses over n, we consider to be true the following set of identities.

$$\begin{aligned} J\left( P_{i}(x) \right)&= \lambda ^{[0]}_{i} P_{i}(x) \; , \, i=0,\ldots ,n+2\, , \end{aligned}$$
(3.7)
$$\begin{aligned} J^{(1)}\left( P_{i}(x)\right)&= \left( \lambda ^{[0]}_{i+1} - \lambda ^{[0]}_{i}\right) P_{i+1}(x) \\ \nonumber \quad + \alpha _{i}&\left( \lambda ^{[0]}_{i-1} - \lambda ^{[0]}_{i} \right) P_{i-1}(x)+ \gamma _{i-1} \left( \lambda ^{[0]}_{i-2}-\lambda ^{[0]}_{i} \right) P_{i-2}(x)\, , \, i=0,\ldots ,n+1\, , \end{aligned}$$
(3.8)
$$\begin{aligned} J^{(2)} \left( P_{i}(x) \right)&= A_{i+2} P_{i+2}(x) + B_{i+1} P_{i+1}(x) + C_{i} P_{i}(x) \\ \nonumber&\quad +D_{i-1} P_{i-1}(x) + F_{i-2} P_{i-2}(x)+ G_{i-3} P_{i-3}(x) + H_{i-4}P_{i-4}(x) , \, i=0,\ldots ,n+1, \end{aligned}$$
(3.9)

where \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(F_{i}\), \(G_{i}\) and \(H_{i}\) are the coefficients defined in (2.38).

Looking at (2.28) knowing that \(J^{(3)}\left( P_{n+1}(x)\right) = \left( a_{2}^{[3]}x^2+a_{1}^{[3]}x+a_{0}^{[3]}\right) P_{n+1}(x)\), and using the third order recurrence relation and the induction hypotheses (3.9), we conclude:

$$\begin{aligned} J^{(2)} \left( P_{n+2}(x) \right)&= A_{n+4} P_{n+4}(x) + B_{n+3} P_{n+3}(x) + C_{n+2} P_{n+2}(x) \\&\quad +D_{n+1} P_{n+1}(x) + F_{n} P_{n}(x)+ G_{n-1} P_{n-1}(x) + H_{n-2}P_{n-2}(x) . \end{aligned}$$

Similarly, when we apply hypotheses (3.8) and (3.9) into (2.29), along with the third order recurrence relation, we deduce:

$$\begin{aligned} J^{(1)}\left( P_{n+2}(x)\right)&= \left( \lambda ^{[0]}_{n+3} - \lambda ^{[0]}_{n+2}\right) P_{n+3}(x) \\ \nonumber \quad + \alpha _{n+2}&\left( \lambda ^{[0]}_{n+1} - \lambda ^{[0]}_{n+2} \right) P_{n+1}(x)+ \gamma _{n+1} \left( \lambda ^{[0]}_{n}-\lambda ^{[0]}_{n+2} \right) P_{n}(x)\, . \end{aligned}$$
(3.10)

Finally, using the hypotheses (3.7), the third order recurrence relation and (3.10), we infer from (2.30) the following:

$$\begin{aligned}&J\left( P_{n+3}(x) \right) = \lambda ^{[0]}_{n+3} P_{n+3}(x)\, , \end{aligned}$$

which completes the induction argument and allow us to assert that

$$\begin{aligned} J\left( P_{n}(x) \right) = \lambda ^{[0]}_{n} P_{n}(x) \end{aligned}$$

for all non-negative values of n. \(\square \)

It is important to remark that, later on, in section 4, it will be proved that the root of polynomial coefficient \(a_{1}(x)\) cannot coincide with the root of \(a_{2}(x)\).

The content of Theorem 3 provides an entire solution written in terms of the polynomial coefficients of the operator J, and generalizes the content of Proposition 2 and Proposition 4 of [23]. The expressions of the recurrence coefficients are now in a reduced format by means of the application of further simplification commands of the computer algebra used.

Moreover, the identities (2.36) and (2.37) allow us to add the following two further differential relations fulfilled by the 2-orthogonal polynomial sequence defined by (3.13)-(3.15), with respect to the operator (3.1), provided that the polynomial coefficient \(a_{3}(x)\) has a double root.

$$\begin{aligned}&\left( a_{1}(x)I+a_{2}(x) D +\frac{1}{2!}a_{3}(x)D^{2} \right) (P_{n}(x)) = \\ \nonumber&\hspace{4cm} a_{1}^{[1]} \Big (P_{n+1}(x) -\alpha _{n}P_{n-1}(x)-2 \gamma _{n-1}P_{n-2}(x) \Big )\, , \end{aligned}$$
(3.11)
$$\begin{aligned}&\left( a_{2}(x)I+ a_{3}(x)D \right) \left( P_{n}(x)\right) = a_{1}^{[1]} \Big ( \left( \beta _{n}-\beta _{n+1}\right) P_{n+1}(x) \Big .\\ \nonumber&\hspace{0.5cm} +2\left( \alpha _{n}-\alpha _{n+1}\right) P_{n}(x)+\left( \alpha _{n}\left( \beta _{n-1}-\beta _{n}\right) + 3\left( \gamma _{n-1}-\gamma _{n}\right) \right) P_{n-1}(x)\\ \nonumber&\hspace{0.5cm} \Big . +2\gamma _{n-1}\left( \beta _{n-2}-\beta _{n}\right) P_{n-2}(x) + \left( -\alpha _{n}\gamma _{n-2}+\alpha _{n-2}\gamma _{n-1}\right) P_{n-3}(x) \Big ) \, , \; n \ge 0. \end{aligned}$$
(3.12)

Example 1

Let us consider a 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) fulfilling

$$\begin{aligned}&J\left( P_{n}(x) \right) = \lambda _{n}^{[0]}P_{n}(x) \end{aligned}$$

where J is defined by (3.1), with polynomial coefficients:

\(a_{3}(x)=\left( x-r\right) ^2\; , \;a_{2}(x)=\left( x-r\right) \) and \(a_{1}(x)=a_{1}^{[1]}x\; \), \(a_{1}^{[1]} \ne 0\).

Taking into account Theorem 3, the recurrence coefficients of the sequence \(\{P_{n}\}_{n \ge 0}\) are the following, provided that \(\gamma _{n} \ne 0\), \(\, n \ge 1 \,.\)

$$\begin{aligned}&\beta _{n} = -\frac{1}{2 a_{1}^{[1]}}(n+1)n \, , \, \;n \ge 0 \, ,\end{aligned}$$
(3.13)
$$\begin{aligned}&\alpha _{n} = \dfrac{n^2\left( -1+n^2+6 r a_{1}^{[1]}\right) }{12 \left( a_{1}^{[1]}\right) ^2} \, , \,\; n \ge 1 \, ,\end{aligned}$$
(3.14)
$$\begin{aligned}&\gamma _{n} =-\dfrac{n(1+n)\left( -1+n^2+6 r a_{1}^{[1]}\right) \left( n(2+n)+6 r a_{1}^{[1]}\right) }{216 \left( a_{1}^{[1]}\right) ^3}\, , \, \;n \ge 1 \, . \end{aligned}$$
(3.15)

Remark 2

In example 1, we notice that if \(r=0\), then \(\gamma _1 = 0\) which contradicts the regularity of the vector functional \(U=(u_{0}, u_{1})^{T}\). In the next section, this aspect will be clarified from the viewpoint of the roots of the polynomial coefficients of J.

Remark 3

Considering the polynomial coefficients of J as in Theorem 3, in particular, assuming \(a_{1}^{[2]} \ne 0\), we know that \(a_{3}(x)\) and \(a_{2}(x)\) have the same root and, thus, performing a translation on the variable x and considering the parameter \(a_{1}^{[1]}\) normalized, we may reduce the operator J to the following layout (cf. [12])

$$\begin{aligned} a_{0}^{[0]}I + \left( x+a_{1}^{[0]}\right) D + a_{1}^{[2]}xD^2+a_{2}^{[3]}x^2D^3\;. \end{aligned}$$

It is important to note that an affine transformation applied to a 2-orthogonal polynomial sequence yields specific modifications in the corresponding recurrence coefficients. Furthermore, taking into consideration Remark 2, we may analyse the parameter \(a_{1}^{[0]}\) in the above operator, with respect to the regularity of the corresponding vector functional.

4 Hahn’s Property and the General 2-Orthogonal Polynomial Eigenfunctions

We say that a 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) is classical in Hahn’s sense, or Hahn-classical, if the corresponding sequence of the derivatives \(Q_{n}(x)=\frac{1}{n+1}DP_{n+1}(x)\,, \; n \ge 0,\) is also a 2-orthogonal polynomial sequence.

In order to analyse the Hahn-classical character of the 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) presented in the previous section, we begin to recall results regarding its dual sequence \(\{u_{n}\}_{n \ge 0}\).

Lemma 8

[22] Let us consider a MPS \(\{ P_{n} \}_{n \ge 0}\) and an operator J of the form (2.13) such that (2.21)-(2.22) hold. Thus,

\({\tilde{P}}_{n}(x) = P_{n}(x),\;\; n \ge 0,\quad \) if and only if \(\quad J\left( u_{n} \right) = \lambda _{n+k}^{[k]} u_{n+k},\;\; n \ge 0\),

where \({\tilde{P}}_{n}(x) = \left( \lambda _{n+k}^{[k]} \right) ^{-1}J\left( P_{n+k}(x) \right) \), \( n \ge 0\).

Furthermore, the forms \(J^{(1)}(u)\), \(J^{(2)}(u)\) and \(J^{(3)}(u)\) have the following definitions which can be deduced from (2.20), for the operator J of third order:

$$\begin{aligned}&J^{(1)}(u)=a_{1}(x)u-D\left( a_{2}(x) u \right) +\frac{1}{2!}D^{2}\left( a_{3}(x) u \right) \end{aligned}$$
(4.1)
$$\begin{aligned}&J^{(2)}(u)=a_{2}(x)u-D\left( a_{3}(x) u \right) \end{aligned}$$
(4.2)
$$\begin{aligned}&J^{(3)}(u)=a_{3}(x)u\end{aligned}$$
(4.3)
$$\begin{aligned}&J^{(m)}(u)=0\, , \; m \ge 4. \end{aligned}$$
(4.4)

Each element of the dual sequence of a 2-orthogonal MPS can be written in terms of the regular functional vector \((u_{0}, u_{1})\). In particular, we have [18] (p. 307)

$$\begin{aligned} u_{2n}&= E_{n}(x) u_{0} +\mathbf { \textsf {a}}_{n-1}(x) u_{1}\, ,\end{aligned}$$
(4.5)
$$\begin{aligned} u_{2n+1}&= \mathbf {\textsf {b}}_{n}(x) u_{0} + V_{n}(x) u_{1} \, ,\; \end{aligned}$$
(4.6)

where \(\deg \left( E_{n}(x) \right) = \deg \left( V_{n}(x) \right) =n\), \( \deg \left( \mathbf { \textsf {a}}_{n}(x) \right) \le n \), \(\deg \left( \mathbf { \textsf {b}}_{n}(x) \right) \le n\),

\(E_{0}(x)=1\,,\, \mathbf { \textsf {a}}_{-1}(x)=0\), \(\mathbf { \textsf {b}}_{0}(x)=0\), \(V_{0}(x)=1\).

These polynomial coefficients fulfil the following recurrence relations [18].

$$\begin{aligned}&E_{1}(x)= \frac{1}{\gamma _{1}} \left( x-\beta _{0}\right) \,; \quad \quad \mathbf { \textsf {a}}_{0}(x)= - \frac{\alpha _{1}}{\gamma _{1}}\,; \\&\alpha _{2n+2}E_{n+1}(x)+E_{n}(x)=\left( x-\beta _{2n+1}\right) \mathbf { \textsf {b}}_{n}(x)-\gamma _{2n+2}\mathbf { \textsf {b}}_{n+1}(x)\,;\\&\gamma _{2n+3}E_{n+2}(x)- \left( x-\beta _{2n+2}\right) E_{n+1}(x)=- \mathbf { \textsf {b}}_{n}(x)- \alpha _{2n+3}\mathbf { \textsf {b}}_{n+1}(x)\,;\\&\\&\gamma _{2n+2}V_{n+1}(x)-\left( x-\beta _{2n+1}\right) V_{n}(x)= - \mathbf {\textsf {a}}_{n-1}(x) - \alpha _{2n+2}\mathbf {\textsf {a}}_{n}(x)\,;\\&\alpha _{2n+3}V_{n+1}(x) + V_{n}(x)= \left( x-\beta _{2n+2}\right) \mathbf {\textsf {a}}_{n}(x)- \gamma _{2n+3}\mathbf {\textsf {a}}_{n+1}(x)\, ; \quad n \ge 0. \end{aligned}$$

Proposition 4

[24, 25] Given an operator J as defined in (2.31) and a 2-orthogonal polynomial sequence so that \(J\left( u_{n} \right) = \lambda _{n}^{[0]} u_{n}\), \( n \ge 0\,,\) with regard to the corresponding dual sequence \(\{u_n\}_{n \ge 0}\), we get the following identities.

$$\begin{aligned}&J^{(1)}(u_{0})=p_{0}(x)u_{0}+p_{1}(x)u_{1}\; \, , \text { with}\\ \nonumber&p_{0}(x)=\gamma _{1} E_{1}(x)\left( \lambda _{0}^{[0]}-\lambda _{2}^{[0]} \right) \, , \quad p_{1}(x)=\gamma _{1}\mathbf {\textsf {a}}_{0}(x) \left( \lambda _{1}^{[0]}-\lambda _{2}^{[0]} \right) \,; \end{aligned}$$
(4.7)
$$\begin{aligned}&J^{(1)}(u_{1})=f_{0}(x)u_{0}+f_{1}(x)u_{1}\;\, \text {with}\\ \nonumber&f_{0}(x)=\gamma _{2} \left( \lambda _{0}^{[0]}-\lambda _{3}^{[0]} \right) \mathbf {\textsf {b}}_{1}(x) +\alpha _{2}E_{1}(x)\left( \lambda _{0}^{[0]}-\lambda _{2}^{[0]} \right) \, , \quad \\ \nonumber&f_{1}(x)=\gamma _{2}\left( \lambda _{1}^{[0]}-\lambda _{3}^{[0]} \right) V_{1}(x) + \alpha _{2}\mathbf {\textsf {a}}_{0}(x)\left( \lambda _{1}^{[0]}-\lambda _{2}^{[0]} \right) \, ; \end{aligned}$$
(4.8)
$$\begin{aligned}&J^{(2)}(u_0) = {\overline{p}}_{0}(x)u_{0}+{\overline{p}}_{1}(x)u_{1}\; \, , \text { with}\\ \nonumber&{\overline{p}}_{0}(x) =C_{0}+ F_{0}E_{1}(x)+ G_{0}\mathbf {\textsf {b}}_{1}(x) + H_{0}E_{2}(x) \, , \\ \nonumber&{\overline{p}}_{1}(x) =D_{0}+F_{0}\mathbf {\textsf {a}}_{0}(x) +G_{0}V_{1}(x)+ H_{0}\mathbf {\textsf {a}}_{1}(x) \, ; \end{aligned}$$
(4.9)
$$\begin{aligned}&J^{(2)}(u_1) = {\overline{f}}_{0}(x)u_{0}+{\overline{f}}_{1}(x)u_{1}\; \, , \text { with}\\ \nonumber&{\overline{f}}_{0}(x)= B_1+D_{1}E_{1}(x)+F_{1}\mathbf {\textsf {b}}_{1}(x) + G_{1}E_{2}(x) + H_{1}\mathbf {\textsf {b}}_{2}(x) \, , \\ \nonumber&{\overline{f}}_{1}(x)= C_1+D_{1}\mathbf {\textsf {a}}_{0}(x)+F_{1}V_{1}(x) + G_{1}\mathbf {\textsf {a}}_{1}(x)+ H_{1}V_{2}(x) \, , \end{aligned}$$
(4.10)

where the above polynomial coefficients \({\overline{p}}_{0}(x)\), \({\overline{p}}_{1}(x)\), \({\overline{f}}_{0}(x)\) and \({\overline{f}}_{1}(x)\) are expressed with the help of the constant sequences (2.38).

In the next Lemma, we review a small set of functional identities [24, 25] that are fulfilled by the fundamental pair of functionals \((u_0,u_1)\) when the identity \(J\left( P_{n} (x) \right) = \lambda _{n}^{[0]} P_{n}(x)\) holds.

Lemma 9

Considering an isomorphism J defined by (2.31) and a 2-orthogonal MPS \(\{P_{n}(x)\}_{n \ge 0}\) such that \(J\left( P_{n} (x) \right) = \lambda _{n}^{[0]} P_{n}(x)\,, \, n \ge 0\), the initial elements of the corresponding dual sequence \(\{u_{n} \}_{n \ge 0}\) fulfil the following three identities:

$$\begin{aligned}&D\left( a_{2}(x)u_{0}\right) = \left( 2p_{0}(x)+ 4a_{1}(x) \right) u_{0}+2p_{1}(x)u_{1}, \end{aligned}$$
(4.11)
$$\begin{aligned}&\frac{1}{2}D^2\left( a_{2}(x)u_{1}\right) -3a_{1}^{[1]}u_{1} = D \Big ( f_{0}(x)u_{0}+ \left( 2a_{1}(x) +f_{1}(x) \right) u_{1} \Big ), \end{aligned}$$
(4.12)
$$\begin{aligned}&D \Big ( {\overline{p}}_{0}(x)u_{0}+{\overline{p}}_{1}(x)u_{1} \Big )+ \left( 2a_{1}(x)+4p_{0}(x) \right) u_{0} + 4p_{1}(x) u_{1} = 0 \,, \end{aligned}$$
(4.13)

where polynomials \(p_{0}(x)\) and \(p_{1}(x)\) are defined in (4.7), the polynomials \(f_{0}(x)\) and \(f_{1}(x)\) are defined in (4.8), and the polynomials \({\overline{p}}_{0}(x)\) and \({\overline{p}}_{1}(x)\) are defined in (4.9) or, equivalently, as issued next in (4.14).

$$\begin{aligned}&{\overline{p}}_{0}(x)=\gamma _{1}\gamma _3\Big ( \left( \lambda _{4}^{[0]}-\lambda _{0}^{[0]} \right) E_{2}(x)+ E'_{2}(x)p_{0}(x)+\mathbf {\textsf {a}}'_{1}(x)f_{0}(x) \Big )\, , \\ \nonumber&{\overline{p}}_{1}(x)=\gamma _{1}\gamma _3 \Big ( \left( \lambda _{4}^{[0]}-\lambda _{1}^{[0]} \right) \mathbf { \textsf {a}}_{1}(x)+ E'_{2}(x)p_{1}(x)+\mathbf {\textsf {a}}'_{1}(x)f_{1}(x) \Big ) \,. \end{aligned}$$
(4.14)

Theorem 5

Let us consider a 2-orthogonal polynomial sequence \(\{P_{n}\}_{n \ge 0}\) fulfilling

$$\begin{aligned}&J\left( P_{n}(x) \right) = \lambda _{n}^{[0]}P_{n}(x) \end{aligned}$$

where J is defined by (3.1), with \(a_{1}^{[2]} \ne 0\).

Then, the roots of \(a_{1}(x)\) and \(a_{2}(x)\) are different and the vector functional \(U=(u_{0}, u_{1})^{T}\) is Hahn-classical fulfilling the following identity.

$$\begin{aligned} D\left( \left[ \begin{array}{cc} \phi _{1,1}(x) &{} 0 \\ \phi _{2,1}(x) &{} \;\; \phi _{2,2}(x) \end{array}\right] \left[ \begin{array}{c} u_0 \\ u_1 \end{array} \right] \right) + \left[ \begin{array}{cc}0 &{} 1 \\ 2E_{1}(x) &{} \;\;2\mathbf {\textsf {a}}_{0}(x) \end{array}\right] \left[ \begin{array}{c} u_0 \\ u_1 \end{array}\right] = \left[ \begin{array}{c} 0 \\ 0 \end{array}\right] \,, \end{aligned}$$
(4.15)

where

$$\begin{aligned}&\phi _{1,1}(x)= -\dfrac{1}{2a_{1}^{[1]}\alpha _{1}}a_{2}(x)\, ,\\&\phi _{2,1}(x)= \dfrac{1}{3a_{1}^{[1]}\gamma _{1}}\left( a_{2}(x)- {\overline{p}}_{0}(x)\right) \, ,\\&\phi _{2,2}(x)= -\dfrac{1}{3a_{1}^{[1]}\gamma _{1}}{\overline{p}}_{1}(x) \, . \end{aligned}$$

Proof

Considering the following polynomial coefficients, as in Theorem 3, in particular, assuming \(a_{1}^{[2]} \ne 0\):

$$\begin{aligned} a_{3}(x)= a_{2}^{[3]}\left( x +\dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \right) ^2\, \, ,\;\; a_{2}(x)= a_{1}^{[2]}\left( x +\dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \right) \, , \end{aligned}$$

we get about the polynomials \(p_{0}(x)\) and \(p_{1}(x)\) indicated in Lemma 9:

$$\begin{aligned}&2p_{0}(x)=-4a_{1}^{[1]}\left( x-\beta _{0} \right) = -4a_{1}^{[1]}\left( x+\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} \right) \,,\\&\text {thus} \; ,\; 2p_{0}(x)+4a_{1}(x)=0\,,\\&2p_{1}(x) = 2a_{1}^{[1]}\alpha _{1}\; , \text {where} \, \alpha _{1}= \dfrac{1}{2a_{1}^{[1]}}\left( -a_{0}^{[2]} + \dfrac{a_{0}^{[1]}a_{1}^{[2]}}{a_{1}^{[1]}} \right) \,,\\&2a_{1}(x) + 4 p_{0}(x) = -6a_{1}^{[1]}\left( x-\beta _{0} \right) = -6a_{1}^{[1]}\left( x+\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} \right) \, . \end{aligned}$$

Taking into account Lemma 7, in what follows we may consider two cases:

either (I) : \(\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} \ne \dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \), thus \( \alpha _{1} \ne 0\), or (II) : \(\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} = \dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \), thus \( \alpha _{1} = 0\).

(I) If \(\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} \ne \dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \), then \(2p_{1}(x) \ne 0\), and we may consider identity (4.11) divided by \(-2p_{1}(x)= -2a_{1}^{[1]}\alpha _{1}\), and also (4.13) divided by \(-3a_{1}^{[1]}\gamma _{1}\) that together allow us to write the following matrix identity

$$\begin{aligned} D\left( \left[ \begin{array}{cc} -\dfrac{1}{2a_{1}^{[1]}\alpha _{1}}a_{2}(x) &{} 0 \\ -\dfrac{1}{3a_{1}^{[1]}\gamma _{1}}{\overline{p}}_{0}(x) &{} \;\;-\dfrac{1}{3a_{1}^{[1]}\gamma _{1}}{\overline{p}}_{1}(x) \end{array}\right] \left[ \begin{array}{c} u_0 \\ u_1 \end{array} \right] \right) + \left[ \begin{array}{cc}0 &{} 1 \\ 2E_{1}(x) &{} \;\;-\dfrac{4\alpha _1}{3\gamma _1} \end{array}\right] \left[ \begin{array}{c} u_0 \\ u_1 \end{array}\right] = \left[ \begin{array}{c} 0 \\ 0 \end{array}\right] \end{aligned}$$

which is equivalent to the identity (4.15) enunciated.

Taking into account the characterization of the classical character of d-orthogonal MPS given in p.182–183 of [7], we then conclude that \(\{P_{n}(x)\}_{n \ge 0}\) is Hahn-classical, that is, it fulfils Hahn’s property. It is worth notice that \(\deg \left( \phi _{1,1}(x)\right) \le 1\), \(\deg \left( \phi _{2,1}(x)\right) \le 1\) and \(\deg \left( \phi _{2,2}(x)\right) \le 1\).

(II) If \(\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} = \dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \), then \(\alpha _1=0\) and \(2p_{1}(x) = 0\). On the other hand,

$$\begin{aligned}&2p_{0}(x)=-4a_{1}^{[1]}\left( x+\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} \right) \,, \text {thus} \; ,\; 2p_{0}(x)+4a_{1}(x)=0\,. \end{aligned}$$

Hence, we get from identity (4.11): \(D\left( a_{2}(x)u_{0}\right) =0\), and thus \(a_{2}(x)u_{0}=0\). Taking into account that \(\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} = \dfrac{a_{0}^{[2]}}{a_{1}^{[2]}} \) and \(\beta _0 = -\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}}\), we may state:

$$\begin{aligned} a_{2}(x)u_{0}=0&\Leftrightarrow a_{1}^{[2]}\left( x+\dfrac{a_{0}^{[1]}}{a_{1}^{[1]}} \right) =0\, , \\&\Leftrightarrow a_{1}^{[2]}P_{1}(x)u_{0}=0\, . \end{aligned}$$

If \(a_{1}^{[2]} \ne 0\), then \(\langle u_{0}, P_{1}(x)P_{\nu }(x) \rangle = 0\), \(\forall \nu \ge 0\), which contradicts the regularity conditions (2.7). In this way, we conclude that such a case is impossible.\(\square \)

5 Other Functional Identities

Let us consider the sequence of the derivatives \(Q_{n}(x)=\frac{1}{n+1}DP_{n+1}(x)\,,\) \( \; n \ge 0,\) of any given MPS \(\{P_{n}\}_{n \ge 0}\) and let us notate the corresponding dual sequence by \(\left( v_{n} \right) _{n \ge 0}\). We know that the dual sequences of \(\{P_{n}\}_{n \ge 0}\) and \(\{Q_{n}\}_{n \ge 0}\) are connected by the following identity [8, 17].

$$\begin{aligned} D\left( v_n\right) =-(n+1)u_{n+1}\, , \, n \ge 0\, \end{aligned}$$
(5.1)

Lemma 10

If \(J\left( u_{n} \right) = \lambda _{n}^{[0]} u_{n},\;\; n \ge 0\), where J is any third order differential operator (2.31), then \(\left( v_{n} \right) _{n \ge 0}\) fulfils:

$$\begin{aligned} \lambda _{n+1}^{[0]}v_n&= -\dfrac{1}{6}D^3\left( a_{3}(x) v_n\right) + D^2\left( \left( \dfrac{1}{6}a_{3}'(x)+\dfrac{1}{2}a_{2}(x) \right) v_n\right) \\ \nonumber&-D\left( \left( \dfrac{1}{2}a_{2}'(x)+a_{1}(x) \right) v_n\right) +\left( a_{1}'(x)+a_{0}(x) \right) v_n\,\; , \, n \ge 0\,. \end{aligned}$$
(5.2)

Proof

The identity \(J\left( u_{n+1} \right) = \lambda _{n+1}^{[0]} u_{n+1}\) reads

$$\begin{aligned} -\dfrac{1}{6}D^3\left( a_3(x)u_{n+1}\right) +\dfrac{1}{2}D^2\left( a_{2}(x)u_{n+1}\right) -D\left( a_{1}(x)u_{n+1}\right) +a_{0}(x)u_{n+1}= \lambda _{n+1}^{[0]} u_{n+1}\; , \end{aligned}$$

where in view of (5.1), we can replace \(u_{n+1}\) by \(-\dfrac{1}{n+1}D\left( v_{n} \right) \) and taking into account (2.10), we get:

$$\begin{aligned}&\dfrac{1}{6}D^4\left( a_3(x)v_{n}\right) -\dfrac{1}{6}D^3\left( a'_3(x)v_{n}\right) -\dfrac{1}{2}D^3\left( a_2(x)v_{n}\right) +\dfrac{1}{2}D^2\left( a'_2(x)v_{n}\right) \\&+ D^2\left( a_1(x)v_{n}\right) -D\left( a'_1(x)v_{n}\right) -a_{0}(x)D\left( v_{n}\right) = -\lambda _{n+1}^{[0]}D\left( v_{n} \right) \; , \end{aligned}$$

hence

$$\begin{aligned}&\dfrac{1}{6}D^3\left( a_3(x)v_{n}\right) -D^2\left( \left( \dfrac{1}{6}a'_3(x)+ \dfrac{1}{2}a_2(x) \right) v_{n}\right) + D\left( \left( \dfrac{1}{2}a'_2(x)+a_{1}(x) \right) v_{n}\right) \\&-\left( a'_1(x) + a_0(x) \right) v_{n} = -\lambda _{n+1}^{[0]}v_{n}\; , \quad \end{aligned}$$

which completes the proof. \(\square \)

On the other hand, from (4.1) we may write:

$$\begin{aligned} J^{(1)}\left( Dv_{n} \right) =a_{1}(x)Dv_{n} -D\left( a_{2}(x) Dv_{n} \right) +\frac{1}{2!}D^{2}\left( a_{3}(x) Dv_{n} \right) \end{aligned}$$

and once more applying (2.10), we get

$$\begin{aligned} J^{(1)}\left( Dv_{n} \right)&=-a'_{1}(x)v_{n} +D\Big (\left( a_{1}(x)+ a'_{2}(x) \Big ) v_{n} \right) \\ \nonumber&-D^{2}\left( \left( a_{2}(x)+\frac{1}{2}a'_{3}(x) \right) v_{n} \right) + \frac{1}{2!}D^{3}\left( a_{3}(x)v_{n}\right) \; , \; n \ge 0\,. \end{aligned}$$
(5.3)

Theorem 6

Given an operator J as defined in (2.31) and a 2-orthogonal polynomial sequence so that \(J\left( u_{n} \right) = \lambda _{n}^{[0]} u_{n}\), \( n \ge 0\,,\) where \(\{u_n\}_{n \ge 0}\) is the corresponding dual sequence, we conclude that

$$\begin{aligned} \left( a_{1}^{[1]}+a_{2}^{[2]}\right) u_1 + D\left( \left( a_{1}(x)+\frac{1}{2}a'_{2}(x)\right) u_{1} \right) =0\;. \end{aligned}$$
(5.4)

Proof

Let us begin to read (5.3) with \(n=0\):

$$\begin{aligned} J^{(1)}\left( Dv_{0} \right)&=-a'_{1}(x)v_{0} +D\Big (\left( a_{1}(x)+ a'_{2}(x) \Big ) v_{0} \right) \\ \nonumber&-D^{2}\left( \left( a_{2}(x)+\frac{1}{2}a'_{3}(x) \right) v_{0} \right) + \frac{1}{2!}D^{3}\left( a_{3}(x)v_{0}\right) \; , \end{aligned}$$
(5.5)

and taking into account that \(u_1 = -Dv_0\) and that \(J^{(1)}\left( u_1 \right) \) can also be expressed as indicated in (4.8), we state:

$$\begin{aligned}&a'_{1}(x)v_{0} -D\Big (\left( a_{1}(x)+ a'_{2}(x) \Big ) v_{0} \right) \\ \nonumber&+D^{2}\left( \left( a_{2}(x)+\frac{1}{2}a'_{3}(x) \right) v_{0} \right) - \frac{1}{2!}D^{3}\left( a_{3}(x)v_{0}\right) =f_{0}(x)u_{0}+f_{1}(x)u_{1}\,. \end{aligned}$$
(5.6)

Besides, from (5.2) we may write:

$$\begin{aligned}&D^3\left( a_{3}(x) v_0\right) = -6\lambda _{1}^{[0]}v_0 + 6D^2\left( \left( \dfrac{1}{6}a_{3}'(x)+\dfrac{1}{2}a_{2}(x) \right) v_0\right) \\&-6D\left( \left( \dfrac{1}{2}a_{2}'(x)+a_{1}(x) \right) v_0\right) +6\left( a_{1}'(x)+a_{0}(x) \right) v_0\,\;, \end{aligned}$$

and replacing the term of \(D^3\left( a_{3}(x) v_0\right) \) in (5.6), we obtain:

$$\begin{aligned}&a_{1}^{[1]}v_0 - D\left( \left( \dfrac{5}{2}a_{2}'(x)+4a_{1}(x) \right) v_0\right) -\dfrac{1}{2}D^2\left( a_{2}(x)v_0\right) =f_{0}(x)u_{0}+f_{1}(x)u_{1}\,, \end{aligned}$$

that by derivation reads

$$\begin{aligned}&-a_{1}^{[1]}u_1 - D^2\left( \left( \dfrac{5}{2}a_{2}'(x)+4a_{1}(x) \right) v_0\right) -\dfrac{1}{2}D^3\left( a_{2}(x)v_0\right) =D\left( f_{0}(x)u_{0}+f_{1}(x)u_{1}\right) \, . \end{aligned}$$
(5.7)

In view of

$$\begin{aligned} D^2\left( a_{2}(x)u_{1}\right) = -D^3\left( a_{2}(x)v_{0}\right) +D^2\left( a'_{2}(x)v_{0}\right) \,, \end{aligned}$$

we conclude now that (4.12) indicates

$$\begin{aligned}&D^2\left( a'_{2}(x)v_{0}\right) -6a_{1}^{[1]}u_{1} - D \Big ( 2f_{0}(x)u_{0}+ \left( 4a_{1}(x) +2f_{1}(x) \right) u_{1} \Big ) = D^3\left( a_{2}(x)v_{0}\right) \,. \end{aligned}$$
(5.8)

Replacing the term of \( D^3\left( a_{2}(x)v_{0}\right) \) of (5.7) with the expression of (5.8), we obtain:

$$\begin{aligned} 2a_{1}^{[1]}u_{1} + 2D \Big ( a_{1}(x)u_{1}\Big ) -D^2\left( \left( 4a_{1}(x)+3 a'_{2}(x)\right) v_{0}\right) =0\,. \end{aligned}$$
(5.9)

Computing \(D^2\left( \left( 4a_{1}(x)+3 a'_{2}(x)\right) v_{0}\right) \), we get

$$\begin{aligned} D^2\left( \left( 4a_{1}(x)+3 a'_{2}(x)\right) v_{0}\right) =-2 \left( 4a'_{1}(x)+3 a''_{2}(x)\right) u_{1}- \left( 4a_{1}(x)+3 a'_{2}(x)\right) Du_{1}, \end{aligned}$$

and finally (5.9) allow us to infer (5.4). \(\square \)

Corollary 1

Given an operator J as defined in (2.31) and a 2-orthogonal polynomial sequence so that \(J\left( u_{n} \right) = \lambda _{n}^{[0]} u_{n}\), \( n \ge 0\,,\) where \(\{u_n\}_{n \ge 0}\) is the corresponding dual sequence, and so that \(\lambda _{2}^{[0]} \ne \lambda _{1}^{[0]}\), we conclude that \(\deg \left( a_{2}(x) \right) \le 1\,.\)

Proof

Let us consider the identity \( J(u_1)= \lambda _{1}^{[0]} u_{1}\) that corresponds to

$$\begin{aligned} \left( a_{0}(x)- \lambda _{1}^{[0]} \right) u_{1} = D\Big (a_{1}(x)u_1-\frac{1}{2}D\left( a_{2}(x)u_1\right) +\frac{1}{6}D^2\left( a_{3}(x)u_1\right) \Big ). \end{aligned}$$

If \(\lambda _{2}^{[0]} \ne \lambda _{1}^{[0]}\), then \(a_{1}^{[1]}+a_{2}^{[2]} \ne 0\), and by (5.4)

$$\begin{aligned} u_1= -\left( a_{1}^{[1]}+a_{2}^{[2]}\right) ^{-1}D\left( \left( a_{1}(x)+\frac{1}{2}a'_{2}(x)\right) u_{1} \right) \;. \end{aligned}$$

Hence, from the identity \( J(u_1)= \lambda _{1}^{[0]} u_{1}\), we get

$$\begin{aligned} a_{1}^{[1]}D\left( \left( a_{1}(x)+\frac{1}{2}a'_{2}(x)\right) u_{1}\right) = \left( a_{1}^{[1]}+a_{2}^{[2]}\right) D\Big (a_{1}(x)u_1-\frac{1}{2}D\left( a_{2}(x)u_1\right) +\frac{1}{6}D^2\left( a_{3}(x)u_1\right) \Big )\,, \end{aligned}$$

thus,

$$\begin{aligned} a_{1}^{[1]}\left( \left( a_{1}(x)+\frac{1}{2}a'_{2}(x)\right) u_{1}\right) = \left( a_{1}^{[1]}+a_{2}^{[2]}\right) \Big (a_{1}(x)u_1-\frac{1}{2}D\left( a_{2}(x)u_1\right) +\frac{1}{6}D^2\left( a_{3}(x)u_1\right) \Big )\,, \end{aligned}$$

that after a few calculations yields:

$$\begin{aligned} \left( \frac{1}{2}a_{1}^{[1]}a_{1}^{[2]}-a_{2}^{[2]}a_{0}^{[1]}\right) u_{1}= \left( a_{1}^{[1]}+a_{2}^{[2]}\right) D\Big (-\frac{1}{2}a_{2}(x)u_1 +\frac{1}{6}D\left( a_{3}(x)u_1\right) \Big ) \,. \end{aligned}$$

Applying again (5.4), we conclude

$$\begin{aligned} \varphi (x) u_{1} = \frac{1}{6}\left( a_{1}^{[1]}+a_{2}^{[2]}\right) ^2 D\left( a_{3}(x)u_1\right) \,, \end{aligned}$$
(5.10)

where

$$\begin{aligned}&\varphi (x) = \left( a_{2}^{[2]}a_{0}^{[1]}-\frac{1}{2}a_{1}^{[1]}a_{1}^{[2]}\right) \left( a_{1}(x)+\frac{1}{2}a'_{2}(x)\right) +\frac{1}{2}\left( a_{1}^{[1]}+a_{2}^{[2]}\right) ^2 a_{2}(x)\,. \end{aligned}$$
(5.11)

Looking at (4.10), we know that

$$\begin{aligned} a_{2}(x)u_1-D\left( a_{3}(x) u_1 \right) = {\overline{f}}_{0}(x)u_{0}+{\overline{f}}_{1}(x)u_{1}\; \,, \end{aligned}$$

where in light of (5.10), corresponds to

$$\begin{aligned} \left( a_{2}(x)-\dfrac{6}{\left( a_{1}^{[1]}+a_{2}^{[2]}\right) ^2}\; \varphi (x) \right) u_{1} = {\overline{f}}_{0}(x)u_{0}+{\overline{f}}_{1}(x)u_{1}. \end{aligned}$$

Applying Lemma 2, we infer

$$\begin{aligned} a_{2}(x)-\dfrac{6}{\left( a_{1}^{[1]}+a_{2}^{[2]}\right) ^2}\; \varphi (x) = {\overline{f}}_{1}(x), \end{aligned}$$

and comparing the leading coefficients of each member, we obtain \(a_{2}^{[2]}=0\). \(\square \)

It is important to emphasise that Lemma 10, Theorem 6 and Corollary 1 are focused on the general third order differential operator J as defined in (2.31).

To conclude this section, let us reanalyse the final identity of section 4 in view of the functional identity of Theorem 6. In other words, let us consider the polynomial coefficients of the operator J as in Theorem 3 and let us assume \(a_{1}^{[2]} \ne 0\) and recall that \(a_{1}^{[1]} \ne 0\). On one hand, we are able to write identity (5.4) as follows

$$\begin{aligned} a_{1}^{[1]}u_1 + D\left( \left( a_{1}(x)+\frac{1}{2}a_{1}^{[2]}\right) u_{1} \right) =0\;, \end{aligned}$$
(5.12)

and on the other hand, identity (4.11) and the posterior information asserts

$$\begin{aligned}&D\left( a_{2}(x)u_{0}\right) = 2a_{1}^{[1]}\alpha _{1}u_{1}, \end{aligned}$$
(5.13)

where \(\alpha _{1}= \dfrac{1}{2a_{1}^{[1]}}\left( -a_{0}^{[2]} + \dfrac{a_{0}^{[1]}a_{1}^{[2]}}{a_{1}^{[1]}} \right) \ne 0\,\) as explained in section 4.

Inserting the expression of the form \(u_1\) given by (5.13) in the identity (5.12), we obtain

$$\begin{aligned} a_{1}^{[1]} a_{2}(x)u_{0} + \left( a_{1}(x)+\frac{1}{2}a_{1}^{[2]} \right) D\left( a_{2}(x)u_{0}\right) = 0\;, \end{aligned}$$

that yields the following functional identity fulfilled by the form \(u_0\), taking into account (2.10).

$$\begin{aligned}&\left( a_{1}(x)+\frac{1}{2}a_{1}^{[2]} \right) a_{2}(x)u_{0} = 0 \end{aligned}$$
(5.14)

From this identity we deduce the linear homogeneous difference equation with constant coefficients fulfilled by the moments of \(u_0\):

$$\begin{aligned} a_{1}^{[1]}a_{1}^{[2]}\left( u_0\right) _{n+2} + \left( a_{1}^{[1]}a_{0}^{[2]} + a_{1}^{[2]}\left( a_{0}^{[1]} + \frac{1}{2}a_{1}^{[2]}\right) \right) \left( u_0\right) _{n+1} + a_{0}^{[2]}\left( a_{0}^{[1]} +\frac{1}{2}a_{1}^{[2]}\right) \left( u_0\right) _{n}=0\;, \end{aligned}$$

that allow us to define explicitly the sequence \(\left( u_0\right) _{n}\), and consequently the sequence \(\left( u_1\right) _{n}\), depending on the parameters involved since the roots of the characteristic polynomial are

$$\begin{aligned} r_1 = -\dfrac{a_{0}^{[1]}+\frac{1}{2}a_{1}^{[2]}}{a_{1}^{[1]}}\; \text{ and } \; r_2 = -\dfrac{a_{0}^{[2]}}{a_{1}^{[2]}}. \end{aligned}$$

Finally, we may add that identity (5.14) indicates also that in this setup, the single form \(u_0\) is not associated to an orthogonal polynomial sequence. With regard to the question of whether or not we can associate an orthogonal polynomial sequence to each element of the regular vector functional \(U=(u_{0}, \ldots , u_{d-1})^{T}\), we refer to the comments provided in [9] (Remarks 1.2) which shed some light on the subject.