The Symmetrical $H_{q}$-Semiclassical Orthogonal Polynomials of Class One

We investigate the quadratic decomposition and duality to classify symmetrical $H_{q}$-semiclassical orthogonal $q$-polynomials of class one where $H_{q}$ is the Hahn's operator. For any canonical situation, the recurrence coefficients, the $q$-analog of the distributional equation of Pearson type, the moments and integral or discrete representations are given.

Other families of semiclassical orthogonal polynomials of class greater than one were discovered by solving functional equations of the type P (x)u = Q(x)v, where P , Q are two polynomials cunningly chosen and u, v two linear forms [19,26,27,34]. For other relevant works in the semiclassical case see [5,23].
In [21], instead of the derivative operator, the q-difference one is used to establish the theory and characterizations of H q -semiclassical orthogonal q-polynomials. Some examples of H qsemiclassical orthogonal q-polynomials are given in [2,13]. The H q -classical case is exhaustively described in [20,32]. Moreover, in [30] the symmetrical D ω -semiclassical orthogonal polynomials of class one are completely described by solving the system of their Laguerre-Freud equations where D w is the Hahn's operator.
So, the aim of this paper is to present the classification of the symmetrical H q -semiclassical orthogonal q-polynomials of class one by investigating the quadratic operator σ, the q-analog of the distributional equation of Pearson type satisfied by the corresponding form and some H qclassical situations (see Tables 1 and 2) in connection with our problem. Among the obtained canonical cases, three are well known: two symmetrical Brenke type (MOPS) [8,9,10] and a symmetrical case of the Al-Salam and Verma (MOPS) [2]. Also, q-analogues of H(µ), G(α, β) and B[ν] appear. In [3,33], the authors have established, up a dilation, a q-analogues of H(µ) and B[ν] using other methods. For any canonical case, we determine the recurrence coefficient, the q-analog of the distributional equation of Pearson type, the moments and a discrete measure or an integral representation.

Preliminary and notations
Let P be the vector space of polynomials with coefficients in C and let P ′ be its topological dual. 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. Moreover, a form (linear functional) u is called symmetric if (u) 2n+1 = 0, n ≥ 0.
We have the well known formula [25] f (x)σu = σ f x 2 u .
Let {B n } n≥0 be a sequence of monic polynomials with deg B n = n, n ≥ 0, the form u is called regular if we can associate with it a sequence of polynomials {B n } n≥0 such that u, B m B n = r n δ n,m , n, m ≥ 0; r n = 0, n ≥ 0. The sequence {B n } n≥0 is then said orthogonal with respect to u. {B n } n≥0 is an (OPS) and it can be supposed (MOPS). The sequence {B n } n≥0 fulfills the recurrence relation When u is regular, {B n } n≥0 is a symmetrical (MOPS) if and only if β n = 0, n ≥ 0.

Some results about the H q -semiclassical character
A form u is called H q -semiclassical when it is regular and there exist two polynomials Φ and Ψ, Φ monic, deg Φ = t ≥ 0, deg Ψ = p ≥ 1 such that The H q -semiclassical character is kept by a dilation [21]. In fact, let {a −n (h a B n )} n≥0 , a = 0; when u satisfies (2.5), then h a −1 u fulfills the q-analog of the distributional equation of Pearson type where Z Φ is the set of zeros of Φ. In particular, when s = 0 the form u is usually called H q -classical (Al-Salam-Carlitz, big q-Laguerre, q-Meixner, Wall, . . . ) [20].

Lemma 1 ([21]
). Let u be a symmetrical H q -semiclassical form of class s satisfying (2.5). The following statements holds i) If s is odd then the polynomial Φ is odd and Ψ is even.
ii) If s is even then the polynomial Φ is even and Ψ is odd.
In the sequel we are going to use some H q -classical forms [20], resumed in Table 1 (canonical cases: 1.1-1.8) and Table 2 (limiting cases: 2.1-2.3). In fact, when q → 1 in results of Table 2, we recover the classical Laguerre L(α), Bessel B(α) and h − 1 is the Jacobi classical form [24].
Moreover in what follows we are going to use the logarithmic function denoted by Log : Log is the principal branch of log and includes ln : R + \{0} −→ R as a special case. Consequently, the principal branch of the square root is

On quadratic decomposition of a symmetrical regular form
Let u be a symmetrical regular form and {B n } n≥0 be its MOPS satisfying (2.3) with β n = 0, n ≥ 0. It is very well known (see [8,25]) that where {P n } n≥0 and {R n } n≥0 are the two MOPS related to the regular form σu and xσu respectively. In fact, [8,25] u is regular ⇔ σu and xσu are regular, u is positive definite ⇔ σu and xσu are positive definite.
Furthermore, taking we get [8,25] and Consequently, Proposition 1. Let u be a symmetrical regular form.
(i) The moments of u are If u is positive definite and σu has the integral representation then, a possible integral representation of u is (2.14) Proof . (i) is a consequence from the definition of the quadratic operator σ. For (ii) taking into account (2.10), (2.11) we get Hence the desired result (2.12) holds. For (iii) consider f ∈ P and let us split up the polynomial f accordingly to its even and odd parts (2.15) Therefore since u is a symmetrical form By (2.13) and according to (2.16), (2.17) we recover the representation in (2.14).
3 Symmetrical H √ q -semiclassical orthogonal polynomials of class one Proof . From the definition of H q we get Therefore, ∀ f ∈ P, Thus the desired result.
2) u is regular for any q ∈ C.

5) we have the following discrete representation
Proof . The results in 1), 2) are obvious from (3.13). For 3), it is clear that u satisfies (3.14).
from which we get that SV(a, q) is of class one because a = 0, a = q − 1 2 , a = q −n−1 , n ≥ 0. The results mentioned in (3.19)-(3.21) are easily obtained from those well known the properties of the little q-Laguerre from (case 1.2 in Table 1) and (2.10)-(2.14).
Remark 2. The regular form SV(q − 1 2 , q) is the discrete √ q-Hermite form which is H √ qclassical [20]. Table 1) that satisfies In accordance of (3.5), (3.6) we get We recognize the Brenke type symmetrical regular form Y(b, q) [8,9,10]. In [13] it is proved that Y(b, q) is H √ q -semiclassical of class one for b = 0, b = √ q, b = q −n , n ≥ 0 satisfying Also in that work, moments, discrete and integral representations are established.
Proposition 5. The symmetrical form u is a H √ q -semiclassical form of class one for b = 0, b = q n+1+α , n ≥ 0, α ∈ R satisfying Moreover, we have the following identities
C. In the case ϕ(x) = x − 1 the q-analogue of Jacobi form (case 2.3 in Table 2), therefore the q-analog of the distributional equation of Pearson type (3.3), (3.4) become with the constraints By Using the above results and the relations we deduce from (2.9) for n ≥ 0 .