An Introduction to the q-Laguerre-Hahn Orthogonal q-Polynomials

Orthogonal q-polynomials associated with q-Laguerre-Hahn form will be studied as a generalization of the q-semiclassical forms via a suitable q-difference equation. The concept of class and a criterion to determinate it will be given. The q-Riccati equation satisfied by the corresponding formal Stieltjes series is obtained. Also, the structure relation is established. Some illustrative examples are highlighted.


Introduction and preliminary results
The concept of the usual Laguerre-Hahn polynomials were extensively studied by several authors [1,2,4,6,8,9,10,15,18]. They constitute a very remarkable family of orthogonal polynomials taking consideration of most of the monic orthogonal polynomials sequences (MOPS) found in literature. In particular, semiclassical orthogonal polynomials are Laguerre-Hahn MOPS [15,20]. The Laguerre-Hahn set of form (linear functional) is invariant under the standard perturbations of forms [2,9,18,20]. It is well known that a usual Laguerre-Hahn polynomial satisfies a fourth order differential equation with polynomials coefficients but the converse remains not proved until now [20]. Discrete Laguerre-Hahn polynomials were studied in [13]. These families are already extensions of discrete semiclassical polynomials [19]. In literature, analysis and characterization of the q-Laguerre-Hahn orthogonal q-polynomials have not been yet presented in a unified way. However, several authors have studied the fourth order q-difference equation related to some examples of q-Laguerre-Hahn orthogonal q-polynomials such as the co-recursive and the rth associated of q-classical polynomials [11,12]. More generally, the fourth order difference equation of Laguerre-Hahn orthogonal on special non-uniform lattices polynomials was established in [4]. For other relevant works in the domain of orthogonal q-polynomials and q-difference equation theory see [3,21] and [5].
So the aim of this contribution is to establish a basic theory of q-Laguerre-Hahn orthogonal qpolynomials. We give some characterization theorems for this case such as the structure relation and the q-Riccati equation. We extend the concept of the class of the usual Laguerre-Hahn forms to the q-Laguerre-Hahn case. Moreover, we show that some standard transformation and perturbation carried out on the q-Laguerre-Hahn forms lead to new q-Laguerre-Hahn forms; the class of the resulting forms is analyzed and some examples are treated.
We denote by P the vector space of the polynomials with coefficients in C and by P ′ its dual space whose elements are forms. The action of u ∈ P ′ on f ∈ P is denoted as u, f . In particular, we denote by (u) n := u, x n , n ≥ 0 the moments of u. A linear operator T : P −→ P has a transpose t T : P ′ −→ P ′ defined by t T u, f = u, T f , u ∈ P ′ , f ∈ P.
For instance, for any form u, any polynomial g and any (a, c) ∈ (C \ {0}) × C, we let H q u, gu, h a u, Du, (x − c) −1 u and δ c , be the forms defined as usually [20] and [16] for the results related to the operator H q H q u, f := − u, H q f , gu, f := u, gf , h a u, f := u, h a f , where for all f ∈ P and q ∈ C := z ∈ C, z = 0, z n = 1, n ≥ 1 [16] ( In particular, this yields to where (u) −1 = 0 and [n] q := q n −1 q−1 , n ≥ 0 [15]. It is obvious that when q → 1, we meet again the derivative D.
For f ∈ P and u ∈ P ′ , the product uf is the polynomial [20] (uf )(x) := u, f i x i . This allows us to define the Cauchy's product of two forms: The product defined as before is commutative [20]. Particularly, the inverse u −1 of u if there exists is defined by uu −1 = δ 0 . The Stieltjes formal series of u ∈ P ′ is defined by A form u is said to be regular whenever there is a sequence of monic polynomials {P n } n≥0 , deg P n = n, n ≥ 0 such that u, P n P m = r n δ n,m with r n = 0 for any n, m ≥ 0. In this case, {P n } n≥0 is called a monic orthogonal polynomials sequence MOPS and it is characterized by the following three-term recurrence relation (Favard's theorem) The shifted MOPS { P n := a −n (h a P n )} n≥0 is then orthogonal with respect to u = h a −1 u and satisfies (1.1) with [20] β n = β n a , γ n+1 = γ n+1 a 2 , n ≥ 0.
Moreover, the form u is said to be normalized if (u) 0 = 1. In this paper, we suppose that any form will be normalized. The form u is said to be positive definite if and only if β n ∈ R and γ n+1 > 0 for all n ≥ 0. When u is regular, {P n } n≥0 is a symmetrical MOPS if and only if β n = 0, n ≥ 0 or equivalently (u) 2n+1 = 0, n ≥ 0.
Given a regular form u and the corresponding MOPS {P n } n≥0 , we define the associated sequence of the first kind P The following well known results (see [16,17,20]) will be needed in the sequel.
are the Hankel determinants.
(uθ 0 f )(x) = a n x n−1 (u) 0 + lower order terms, (1.16) Definition 1. A form u is called q-Laguerre-Hahn when it is regular and satisfies the qdifference equation where Φ, Ψ, B are polynomials, with Φ monic. The corresponding orthogonal sequence {P n } n≥0 is called q-Laguerre-Hahn MOPS.
Remark 1. When B = 0 and the form u is regular then u is q-semiclassical [17]. When u is regular and not q-semiclassical then u is called a strict q-Laguerre-Hahn form. According to (1.9) and (1.11), the above equation becomes Then ∆ = Ω = 0 because the form u is regular and not q-semiclassical.
Lemma 4. Consider the sequence { P n } n≥0 obtained by shifting P n , i.e. P n (x) = a −n P n (ax), n ≥ 0, a = 0. When u satisfies (1.17), then u = h a −1 u fulfills the q-difference equation from (1.7) and (1.9). Moreover, by virtue of (1.7) an other time we get

Equation (1.17) becomes
Hence the desired result.
Proposition 1. The number s is an integer positive or zero. In other words, if p = 0, then d ≥ 2 or if 0 ≤ d ≤ 1, then necessarily p ≥ 1.
Proof . Let us show that in case s = −1, the form u is not regular, which is a contradiction. Indeed, when s = −1, we have with c 1 = 1 or c 1 = 0 and c 0 = 1, and where a 0 = 0. The condition H q (Φu) + Ψu + B x −1 u(h q u) , x n = 0, 0 ≤ n ≤ 4 gives successively On the other hand, let us consider the Hankel determinant With (2.6), we get ∆ 2 = 0. Contradiction.

Proposition 2. Let u be a strict q-Laguerre-Hahn form satisfying
and Then, there exist two polynomials Ψ and B such that From the fact that u is a strict q-Laguerre-Hahn form and by virtue of Lemma 3 we get Thus, there exist two polynomials Ψ and B such that Then, formulas (2.7), (2.8) become (2.14) But the polynomials h q −1 Φ 1 and h q −1 Φ 2 are also co-prime. Using the Bezout identity, there exist two polynomials A 1 and A 2 such that Consequently, the operation A 1 ×(2.14 i=1 )+A 2 ×(2.14 i=2 ) leads to (2.9). With (2.11) and (2.13) it is easy to prove (2.10).
Proof . If s 1 = s 2 in (2.9), (2.10) and s 1 = s 2 = s = min (u), then t 1 = t = t 2 . Consequently, Then, it's necessary to give a criterion which allows us to simplify the class. For this, let us recall the following lemma: Lemma 5. Consider u a regular form, Φ, Ψ and B three polynomials, Φ monic. For any zero c of Φ, denoting The following statements are equivalent: Proof . The proof is obtained straightforwardly by using the relations in (1.2) and in (2.1).
. On account of (2.15) we have Therefore, The condition (2.17) is necessary. Let us suppose that c fulfils the conditions Then on account of Lemma 5 (2.16) becomes The condition (2.17) is sufficient. Let us suppose u to be of class s < s. There exist three where d := max( t, r). By Proposition 2, it exists a polynomial χ such that Since s < s hence deg χ ≥ 1. Let c be a zero of χ : χ(x) = (x − c)χ c (x). On account of (1.10) we have Thus r cq = 0 and b cq = 0. Moreover, with (1.8) we have This is contradictory with (2.17). Consequently, s = s, Φ = Φ, Ψ = Ψ and B = B.

Remark 2.
When q −→ 1 we recover again the criterion which allows us to simplify a usual Laguerre-Hahn form [6].
Remark 3. When B = 0 and s = 0, the form u is usually called q-classical [16]. When B = 0 and s = 1, the symmetrical q-semiclassical orthogonal q-polynomials of class one are exhaustively described in [14]. (ii) If s is even, then the polynomials Φ and B are even and Ψ is odd.
Proof . Writing

then (1.17) becomes
Denoting Now, with the fact that u is a symmetrical form then uh q u is also a symmetrical form. Indeed, Thus (2.21) gives and 3 Dif ferent characterizations of q-Laguerre-Hahn forms One of the most important characterizations of the q-Laguerre-Hahn forms is given in terms of a non homogeneous second order q-difference equation so called q-Riccati equation fulfilled by its formal Stieltjes series. See also [6,8,10,15] for the usual case and [13] for the discrete one. (b) The Stieljes formal series S(u) satisfies the q-Riccati equation where Φ and B are polynomials defined in (1.17) and Proof . (a) ⇒ (b). Suppose that (a) is satisfied, then there exist three polynomials Φ (monic), Ψ and B such that H q (Φu)+Ψu+B(x −1 uh q u) = 0. From (1.11) the above q-difference equation becomes From definition of S(u) and the linearity of S we obtain Moreover, The previous relation gives (3.1) with (3.2). (b) ⇒ (a). Let u ∈ P ′ regular with its formal Stieltjes series S(u) satisfying (3.1). Likewise as in the previous implication, formula (3.1) leads to According to (3.2) and (1.12) we deduce that We are going to give the criterion which allows us to simplify the class of q-Laguerre-Hahn form in terms of the coefficients corresponding to the previous characterization. where Z Φ is the set of roots of Φ with Proof . By comparing (2.17) and (3.5), it is enough to prove the following equalities Indeed, on account of (3.2), the definition of the polynomial uf , the definition of the product form uv and (1.8) we have Moreover, Thus (2.17) is equivalent to (3.5). To prove (3.6), according to the definition of the class we may write • If deg Ψ = max deg B − 1, deg Φ − 1 , on account of (3.2) and (3.7) we get the following implications • An other important characterization of the q-Laguerre-Hahn forms is the structure relation. See also [6,15] for the usual case and [13] for the discrete one.
Proposition 8. Let u be a regular form and {P n } n≥0 be its MOPS. The following statements are equivalent: (i) u is a q-Laguerre-Hahn form satisfying (1.17).
which is a polynomial of degree at most n + d. Then, there exists a sequence of complex numbers {λ n,ν } n≥0, 0≤ν≤n+d such that Multiplying both sides of (3.9) by P m , 0 ≤ m ≤ n + d and applying u we get u, ΦP m (H q P n+1 ) − h q u, B(h q −1 P m )(uθ 0 P n+1 ) = λ n,m u, P 2 m , n ≥ 0, 0 ≤ m ≤ n + d.
From definitions and the hypothesis of (ii) we may write successively From assumption of orthogonality of {P n } n≥0 with respect to u we get v, P n = 0, n ≥ s + 2.
In order to get v, P n = 0, for any n ≥ 0, we shall choose a i with i = 0, 1, . . . , s + 1, such that v, P i = 0, for i = 0, 1, . . . , s + 1. These coefficients a i are determined in a unique way. Thus, we have deduced the existence of polynomial Ψ(x) = s+1 i=0 a i x i such that v, P n = 0, for any n ≥ 0. This leads to H q (Φu) + Ψu + B(x −1 uh q u) = 0 and the point (i) is then proved.

The co-recursive of a q-Laguerre-Hahn form
Let µ be a complex number, u a regular form and {P n } n≥0 be its corresponding MOPS satisfying (1.1). We define the co-recursive P n (x), n ≥ 0.
Denoting by u [µ] its corresponding regular form. It is well known that [20, equation (4.14)] Proposition 9. If u is a q-Laguerre-Hahn form of class s, then u [µ] is a q-Laguerre-Hahn form of the same class s.
Replacing the above results in (3.1) the q-Riccati equation becomes Therefore the q-Riccati equation satisfied by S(u [µ] ) where As a consequence, the regular form u [µ] fulfils the following q-difference equation We suppose that the q-Riccati equation (

The associated of a q-Laguerre-Hahn form
Let u be a regular form and {P n } n≥0 its corresponding MOPS satisfying (1.1). The associated sequence of the first kind P (1) n n≥0 of {P n } n≥0 satisfies the following three-term recurrence relation [20] P (1) Denoting by u (1) its corresponding regular form. Proof . We assume that the formal Stieltjes function S(u) of u satisfies (3.1). The relationship between S(u (1) ) and S(u) is [20, equation (4.7)] Consequently, From definitions and by virtue of (4.7) we have Substituting in (3.1) the q-Riccati equation becomes Equivalently Therefore the q-Riccati equation satisfied by S(u (1) ) where Likewise, it is straightforward to prove that the class of u (1) is also s.