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

Abstract

Abstract. 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.

q-Chebyshev polynomials and its q-classical and q-Laguerre-Hahn character

“…The extension of the H q -semiclassical forms are the H q -Laguerre-Hahn forms defined as following [6] Definition 2.4. A regular form w is called H q -Laguerre-Hahn when it satisfies the q-difference equation…”

“…Let us introduce the integer s(φ, ψ, B) = max deg(ψ)−1, max(deg(φ), deg(B))− 2 . Then s = min s(φ, ψ) where the minimum is taken over all the pairs (φ, ψ, B) occurring in (2.16) is called the class of w. By extension, the integer s is also the class of {P n } n≥0 [6].…”

“…Proposition 2.9. [6] Let w be a regular form. The following stataments are equivalent: (a) w is q-Laguerre-Hahn form satisfying (2.16).…”

“…Proposition 2.11. [6] A regular form w q-Laguerre-Hahn form satisfying (2.18) is of class s if and only if…”