On the Finite Orthogonality of q -Pseudo-Jacobi Polynomials

: Using the Sturm–Liouville theory in q -difference spaces, we prove the ﬁnite orthogonality of q -Pseudo Jacobi polynomials. Their norm square values are then explicitly computed by means of the Favard theorem.

It has been acknowledged that the theory of q-special functions is essentially based on the relation Hence, a basic number in q-calculus is defined as There is a q-analogue of the Pochhammer symbol (3) (called q-shifted factorial) as Moreover we have (1 − aq k ) for 0 < |q| < 1, and (a 1 , a 2 , ..., a m ; q) ∞ = (a 1 ; q) ∞ (a 2 ; q) ∞ ...(a m ; q) ∞ .
There exist several q-analogues of classical hypergeometric orthogonal polynomials that are known as basic hypergeometric orthogonal polynomials [3].
On the other side, if we set c = 0, a = q α and b = q β in (8) and then let q → 1, we find the Jacobi polynomials (2) as . Moreover, by referring to (8), one can define another family of big q-Jacobi polynomials [13] with four free parameters as . Because a particular case of Jacobi polynomials (5) are called the pseudo Jacobi polynomials, it is reasonable to similarly consider a special case of big q-Jacobi polynomials preserving the limit relation as q → 1. This means that the q-pseudo Jacobi polynomials will be derived by substituting in a special case of the polynomials (8) as Therefore, the q-pseudo Jacobi polynomials are defined as The main aim of this paper is to apply a q-Sturm-Liouville theorem in order to obtain a finite orthogonality for the real polynomials (11) on (−∞, ∞), which is a new contribution in the literature.
A regular Sturm-Liouville problem of continuous type is a boundary value problem of the form which is defined on an open interval, say (γ 1 , γ 2 ) with the boundary conditions where α 1 , α 2 and β 1 , β 2 are constant numbers and K(x), and w(x) in (12) are to be assumed continuous functions for x ∈ [γ 1 , γ 2 ]. The function w(x) is called the weight or density function. Let y n and y m be two eigenfunctions of Equation (12). According to the Sturm-Liouville theory [14], they have an orthogonality property with respect to the weight function w(x) under the given condition (13), so that we have in which δ m,n = 0 (n = m), 1 (n = m).
There are generally two types of orthogonality for relation (14), i.e. infinitely orthogonality and finitely orthogonality. In the finite case, one has to impose some constraints on n, while in the infinite case, n is free up to infinity [4].
By referring to the differential Equation (6), it is proved in [4] that where Γ(.) is the well-known gamma function. Similarly, q-orthogonal functions can be solutions of a q-Sturm-Liouville problem in the form [15] D q K(x; q)D q y n (x; q) + λ n,q w(x; q)y n (x; q) = 0, (K(x; q) > 0, w(x; q) > 0), where and (15) satisfies a set of boundary conditions like (13). This means that if y n (x; q) and y m (x; q) are two eigenfunctions of the q-difference Equation (15), they are orthogonal with respect to a weight function w(x; q) on a discrete set [16]. Let ϕ(x) and ψ(x) be two polynomials of degree at most 2 and 1, respectively, as If {y n (x; q)} n is a sequence of polynomials that satisfies the q-difference equation [3] ϕ(x)D 2 q y n (x; q) + ψ(x)D q y n (x; q) + λ n,q y n (qx; q) = 0, where λ n,q ∈ C and q ∈ R \ {−1, 0, 1}, then the following orthogonality relation holds and w(x; q) is a solution of the Pearson q-difference equation Note that w(x; q) is assumed to be positive and w(q −1 x; q)ϕ(q −2 x)x k for k ∈ N must vanish at IfP n (x) = x n + · · · is a monic solution of Equation (16), the eigenvalue λ n,q is explicitly derived as The q-integral as an inverse of the q-difference operator [3,17,18] is defined as provided that the series converges absolutely. Furthermore, we have

Theorem 1. Let {J
(u,v) n (x; q)} n defined in (11) be a sequence of polynomials that satisfies the q-difference Equation (18). Subsequently, we have where N < u − 1 2 for N = max{m, n} and the positive function w (u,v) (x; q) is a solution of the Pearson-type q-difference equation which is equivalent to Proof. First, according to [3] and referring to (7) it is not difficult to verify that is a solution of Equation (19). (18) is written in the self-adjoint form