Turán Inequalities for Symmetric Orthogonal Polynomials

A method is outlined to express a Turán determinant of solutions of a three term recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Turán determinants of symmetric Pollaczek polynomials, Lommel polynomials and q-Bessel functions

Turán Inequalities for Symmetric Orthogonal Polynomials

A method is outlined to express a Turán determinant of solutions of a three term recurrence relation as a weighted sum of squares. This method is shown to imply the positivity of Turán determinants of symmetric Pollaczek polynomials, Lommel polynomials and q-Bessel functions

Basic Analog Of Fourier Series On A q-Quadratic Grid

. We prove orthogonality relations for some analogs of trigonometric functions on a q-quadratic grid and introduce the corresponding q-Fourier series. We also discuss several other properties of this basic trigonometric system and the q-Fourier series. 1. Introduction A periodic function with period 2l, f(x + 2l) = f(x); (1.1) can be represented as the Fourier series, f(x) = a 0 + 1 X n=1 i a n cos ßn l x + b n sin ßn l x j ; (1.2) where a 0 = 1 2l Z l \Gammal f(x) dx; (1.3) a n = 1 l Z l \Gammal f(x) cos ßn l x dx; (1.4) b n = 1 l Z l \Gammal f(x) sin ßn l x dx: (1.5) For convergence conditions of (1.2) see, for example, [1], [28], and [30]. The formulas (1.3)--(1.5) for the coefficients of the Fourier series are consequences of the orthogonality relations for trigonometric functions Z l \Gammal cos nßx l cos mßx l dx = 0; m 6= n; (1.6) Z l \Gammal sin nßx l sin mßx l dx = 0; m 6= n; (1.7) Date: May 21, 1997. 1991 Mathematics Subject Classification. Pri..