Μελέτη των ριζών των associated ορθογωνίων q-πολυωνύμων

Abstract

In this thesis, we study the monotonicity properties and the convexity of the zeros of some families of associated orthogonal q-polynomials. Also, we calculate the Newton sum rules of these zeros. For the study of the monotonicity of the zeros, we use a functional analytic method based on the three terms recurrence relations satisfied by the associated orthogonal q-polynomials under consideration. This method was introduced by Ifantis and Siafarikas in [55] for the classical orthogonal polynomials and in [56, 111] for the corresponding associated orthogonal polynomials. Moreover, in order to calculate the Newton sum rules of the zeros of the associated orthogonal q-polynomials we use a functional analytic method, introduced in [52] for the calculation of the Newton sum rules of the zeros of the scaled co-recursive associated orthogonal polynomials. In chapter 3 we study the monotonicity and we give differential inequalities for the largest zero of some associated orthogonal q-polynomials. More precisely, we give monotonicity properties and inequalities for the largest zero of the associated q-Pollaczek polynomials which is a generalization of the q-Pollaczek polynomials introduced in [58] and [17]. Special cases of the associated q-Pollaczek polynomials are the associated continuous q-Ultraspherical polynomials and the associated q-Hermite polynomials for which we can immediately obtain the corresponding results. We also study the monotonicity and we give inequalities for the zeros of the associated continuous q-Jacobi polynomials, the associated q-Laguerre polynomials, the associated Al-Salam-Carlitz II polynomials and the associated q-Meixner polynomials, special case of which are the associated q-Charlier polynomials. The above polynomials are generalizations of the corresponding polynomials that appear in the q-analogue of the Askey scheme. Finally, we study the monotonicity and we give inequalities for the largest zero of the q-Lommel polynomials introduced in [57] in association with the Jackson q-Bessel function J(2) _ (x; q) [70]. Corresponding results for the first zero of J(2) _ (x; q) are also given. By taking the limit q ! 1? and with appropriate changes of the parameters, we obtain analogous results for the largest zero of some families of associated orthogonal polynomials. In chapter 4, we obtain differential inequalities that give information about the monotonicity of the lowest zero of associated orthogonal q-polynomials for the following families of q-polynomials: Askey-Wilson, continuous dual q-Hahn, Al-Salam-Chihara, continuous big q-Hermite, continuous q-Jacobi, continuous q-Laguerre, big q-Jacobi, big q-Laquerre and big q-Legendre. Also, from these inequalities we derive lower bounds for the corresponding interval of orthogonality in some cases where it is unknown. Taking the limit q ! 1?, these results are translated to some families of associated classical orthogonal polynomials, giving new information about Wilson, continuous dual Hahn, Jacobi, Laguerre and Legendre polynomials. In chapter 5, we prove the convexity of the largest zero of the q-Lommel and the associated q-Laguerre polynomials as well as the convexity of products of certain functions with the largest zero of the associated q-Laguerre polynomials and associated Al-Salam-Carlitz II polynomials. Moreover, taking the limit q ! 1? as a consequence of our results concerning the associated q-Laguerre polynomials, we obtain a recent result regarding the convexity of the function 1 _ + 1 xn,1(_), where xn,1(_) is the largest zero of the classical Laguerre L_ n (x) polynomials. Finally, in chapter 6 we give the explicit expressions for the Newton sum rules N?1 Pn=0 xk n(c|q), k = 1, 2, 3, 4 of the zeros of some families of associated orthogonal q-polynomials. We mention that by use of this method it is possible, although tedious, to calculate the Newton sum rules for any desired k. More precisely, we determine the Newton sum rules for the zeros of the associated q-Laguerre, associated continuous q-Ultraspherical, associated continuous q-Legendre, associated continuous q-Hermite, q-Lommel, associated Al-Salam-Carlitz I, II and associated q-Meixner polynomials. From these, if we let q ! 1?, we obtain as particular cases the Newton sum rules for the zeros of the associated Laguerre, associated Charlier, associated Ultraspherical, associated Legendre and associated Hermite polynomials found recently [3, 41, 52, 68, 92, 93]. Also, from the Newton sum rules for the zeros of the q-Lommel polynomials, we obtain the corresponding results for the zeros of q-Bessel functions and from them the known Rayleigh sums [118] for the zeros of Bessel functions. The above results unify, generalize and improve previously known results.