A family of symmetric second degree semiclassical forms of class s = 2

A regular form (linear functional) u is called semiclassical, if there exist two nonzero polynomials \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Psi}$$\end{document} such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${( \Phi u )^{\prime} + \Psi u = 0}$$\end{document} with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi}$$\end{document} monic and deg \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Psi > 0}$$\end{document}. Such a form is said to be of second degree if there are polynomials B, C and D such that its Stieltjes function S(u) satisfies BS2(u) + CS(u) + D = 0. Recently, all the symmetric second degree semiclassical forms of class s ≤ 1 were determined. In this paper, by means of the quadratic decomposition, we determine all the symmetric semiclassical forms of class s = 2, which are also of second degree when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Phi}$$\end{document} vanishes at zero. These forms generalize those of class s = 1.


Introduction and basic background
Second degree forms have been introduced since 1995 [13]. These forms are characterized by the fact that their formal Stieltjes function S(u) satisfies a quadratic equation B S 2 (u) + C S(u) + D = 0 where B = 0 and C are polynomials and D is a polynomial defined in terms of the previous ones. They have been studied in [7,16] and [17] in the framework of the orthogonality on several intervals. Later on, in [12] and [13] an algebraic approach to such second degree forms as an extension of the Tchebychev forms is given. Notice that every second degree form is semiclassical, i.e., there exist two polynomials (x) and (x), where (x) is monic and deg > 0, such that ( (x)u) + (x)u = 0 [11,13]. In [3], the authors determine all the classical forms (i.e., semiclassical of class s = 0) which are of second degree. Hermite, Laguerre and Bessel are not of second degree. Only Jacobi forms which satisfy a certain condition possess this property. Later on, in [2], Beghdadi determines all the symmetric second degree semiclassical forms of class s = 1.
The aim of this work is to approach the problem of determining all the symmetric semiclassical forms of class s = 2 which are of second degree when (0) = 0. The first section is devoted to the preliminary results and notations used in the sequel. In the second section, we prove that a symmetric semiclassical form u is a second degree if and only if its odd part xσ u is also second degree form. Using this result, we give all the forms which we look for. Three canonical cases for the polynomial arise: (x) = x 2 , (x) = x 4 and (x) = x 2 (x 2 − 1). As it turned out, we obtained explicitly a family of nonsymmetric second degree semiclassical forms of class s = 1 which generalize the classical ones.
In the sequel, we will recall some basic definitions and results. The field of complex numbers is denoted by C. The vector space of polynomials with coefficients in C is represented as P and its dual space is represented as P . We denote by u, f the effect of u ∈ P on f ∈ P. In particular, we denote by (u) n := u, x n , n ≥ 0, the moments of u. For any linear form u, any polynomial h, let Du = u and hu be the forms defined by duality: (1) We recall the definition of right-multiplication of a form by a polynomial: By duality, we obtain the Cauchy's product of two forms: Consequently, We define [14] with We introduce the operator σ : P −→ P defined by (σ f )(x) = f (x 2 ) for all f ∈ P. By transposition, we define σ u: Consequently, (σ u) n = (u) 2n . We will also use the so-called formal Stieltjes function associated with u ∈ P that is defined by The following auxiliary results will be used in the sequel [14,15].
The form u is called regular if there exists a polynomial sequence {B n } n≥0 , deg B n = n, such that u, B n B m = r n δ nm , r n = 0, n ≥ 0.
In this case {B n } n≥0 is said to be orthogonal with respect to u. It satisfies the recurrence relation (see, for instance, the monograph by Chihara [4]) The regularity of u means that we must have γ n+1 = 0, n ≥ 0.
Definition 1.2 [13] The form u is called a second degree form if it is regular and if there exist two polynomials B and C such that where D is a polynomial depending on B, C, and u given by The regularity of u means that we must have B = 0; C 2 − 4B D = 0 and D = 0.
The following expressions are equivalent to (16), [13]: In the sequel, we shall suppose B to be monic. The polynomials B and C, given in (16) or by (18), are not unique, because B and C can be multiplied by an arbitrary polynomial. If in (16) the polynomials B , C and D are coprime, then the pair (B, C) is called a primitive pair. The primitive pair is unique. Let us recall that a form u is called semiclassical when it is regular and there exist two polynomials and , where (x) is monic and deg( ) ≥ 1, such that The class of the semiclassical form v is s = max(deg − 1, deg − 2) if and only if the following condition is satisfied where c goes over the zeros set of [14]. When s = 0, u is called a classical form. As a result, if u is a semiclassical form of class s satisfying (19), then the shifted formû = (h a −1 •τ −b )u, a ∈ C * , b ∈ C is of class s satisfying the equation where, for each polynomial f A second degree form u is a semiclassical form and satisfies (19), with [13] k where k is a normalization factor.
The second degree character is kept by shifting. Indeed, if u is a second degree form satisfying (18), then u is also second degree form [13]. It satisfieŝ We finish this section by recalling this important result.

Theorem 1.4 [3] Among the classical forms, only the Jacobi forms
2 Symmetric second degree semiclassical forms
In addition, the sequence {B n } n≥0 has the following quadratic decomposition The sequences {P n } n≥0 and {R n } n≥0 are respectively orthogonal with respect to σ u and xσ u. We have for instance: with We have the following characterisations.

Proposition 2.1 [2]
The even part σ u of a symmetric second degree form u is also second degree form.

Proposition 2.2 Let u be a regular and symmetric form. The following statements are equivalent:
(a) u is a second degree form (b) The odd part xσ u of u is a second degree form.
Hence u is also a second degree form.
Using Proposition 2.1, Beghdadi gives all the symmetric second degree semiclassical forms of class s = 1: Among the symmetric semiclassical forms of class s = 1, only the forms denoted by The form I = I(k − 1 2 , l − 1 2 ) possesses the following representation [2]: Remark Unfortunately, we are not able to determine all the symmetric second degree semiclassical forms of class s = 2 by Proposition 2.1, especially because σ u is among the second degree semiclassical forms of class s = 1 which are unknown.
2.2 Symmetric second degree semiclassical forms of class s = 2: case (0) = 0 Let us begin with an example V among the symmetric forms which is a second degree semiclassical form of class s = 2 satisfying (19) with (0) = 0. This example is given in [1]. The form V satisfies (16) with and (19) with The corresponding MOPS of V satisfies (15) with Now, we state the following result which is essential for this work.

Proposition 2.4 [2] Let u be a symmetric semiclassical form of class s, satisfying (19). If s is even then is even and is odd. If s is odd then is odd and is even.
In the sequel, we suppose s = 2, u is symmetric, and (0) = 0. Then, according to the above proposition, u satisfies (19) with Then, using the fact that is monic and the semiclassical character is kept by shifting, we distinguish three canonical cases for : According to Lemma 1.3, this case is excluded because s = 2 = deg − 2. Second case: (x) = x 4 Let (x) = a 3 x 3 + a 1 x. After multiplying (19) by x, applying the operator σ and using (11)-(12), we obtain Then xσ u = γ 1 B(α) where B(α) is the classical Bessel form with a 3 = −4α + 1 and a 1 = −4. Recall that the form B(α) satisfies (19) with Since B(α) is not a second degree form [3], according to Proposition 2.2, we conclude that u is not a second degree form. Third case: This case is mentioned in [6] and [18], when the authors gave all the symmetric semiclassical forms of class s = 2 with (x) = x 2 (x 2 − 1). These forms satisfy Taking into account [18], we have , n ≥ 0, , n ≥ 0.
The regularity condition is In the sequel, we denote by L(α, β, λ) the form u which satisfies (40).
Proof After multiplying (40) by x, applying the operator σ and using (11)-(12), we obtain Let us make the suitable shift for (xσ u) Using (22), (xσ u) satisfies (21) witĥ where J (a, b) is the classical Jacobi form with Pearson equation According to Theorem 1.4, Proposition 2.2 and the fact that the shifted form of a second degree form is also second degree form, we obtain: L(α, β, λ) is a second degree semiclassical form of class s = 2 if and only if Let us now give the polynomial coefficients B, C and D of (16) corresponding to these forms. For this, we need the following lemmas.

Lemma 2.6 [3]
Let u and v be two regular forms satisfying the following relation: where M(x) and N (x) are two polynomials. If u is a second degree form satisfying (16), then v is also a second degree form and satisfies (47) Lemma 2. 7 We have where μ is the normalization factor.

Integral representation
The form u = L(α, β, λ) has the following representation [6,15] From Theorem 2.5, we deduce the following: A symmetric semiclassical form of class s = 2 satisfying (19) with (0) = 0 is a second degree form and positive definite if the weight function has the following expression: where Y is the characteristic function of R + .
The case p = 0 and q = 0 is V. This form is not positive definite, and has the integral representation [1]
In a very interesting work [5], J. Charris, G. Salas and V. Silva studied this sequence of orthogonal polynomials. This sequence is a particular case of a more general sequence considered in Example 1 presented in [10]. According to Theorem 2.5 we deduce that it is a second degree form.
. This means that the second degree forms u = L( p − 1 2 , q − 1 2 , λ) generalize the symmetric second degree forms of class s = 1.

The study of
In this part, the focus will be put on σ u: the even part of u = L( p − 1 2 , q − 1 2 , λ), provided p + q ≥ 0, p, q ∈ Z.
: the classical second degree forms. Indeed, in this case, we necessarily have p + q = 0. Then, for ( p, q) = (k, l + 1), we obtain the statement of Theorem 1.4.