We study some properties of the Askey-Wilson polynomials (AWP) when q is a primitive Nth root of unity. For general four-parameter AWP, zeros of the Nth polynomial and the orthogonality measure are found explicitly. Special subclasses of the AWP, e.g., the continuous q-Jacobi and big q-Jacobi polynomials, are considered in detail. A set of discrete weight functions positive on a real interval i...

Nonlinear optimal control and nonlinear H 1 control are two of the most signiicant paradigms in nonlinear systems theory. Unfortunately , these problems require the solution of Hamilton-Jacobi equations which are extremely diicult to solve in practice. To make matters worst, approximation techniques for these equations are inherently prone to the so-called \curse of dimensionality." While there...

If ~q1, . . . , ~qm : Z → Z are polynomials with zero constant terms and E ⊂ Z has positive upper Banach density, then we show that the set E ∩ (E − ~q1(p− 1))∩ . . .∩(E−~qm(p−1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results ...

We introduce the concept of D-operators associated to a sequence of polynomials (pn)n and an algebra A of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family (pn)n of ...

We study the combinatorics of a continued fraction formula due to Wall. We also derive the orthogonality of little q-Jacobi polynomials from this formula, as Wall did for little q-Laguerre polynomials.

Let X represent either a space C[−1,1] or L α,β(w), 1 ≤ p < ∞, of functions on [−1,1]. It is well known that X are Banach spaces under the sup and the p-norms, respectively. We prove that there exist the best possible normalized Banach subspaces Xα,β of X such that the system of Jacobi polynomials is densely spread on these, or, in other words, each f ∈ Xα,β can be represented by a linear combi...

Binary Sidel’nikov-Lempel-Cohn-Eastman sequences (or SLCE sequences) over F2 have even period and almost perfect autocorrelation. However, the evaluation of the linear complexity of these sequences is really difficult. In this paper, we continue the study of [1]. We first express the multiple roots of character polynomials of SLCE sequences into certain kinds of Jacobi sums. Then by making use ...

This paper deals with the eigenvalue singular problem of the spectral type differential equations of the fourth-order self-adjoint differential operators 1 ð1 xÞð1þ xÞ d dx ð1 xÞð1þ xÞ d y dx " # ly 1⁄4 0; yð 1Þ; yð1Þ finite where, p P 1; q P 1. It has been reported in the literature that spectral type differential equations above have general solutions in terms of hypergeometric functions. In ...