Frederick Ira Moxley III
$q^{-1}$-Orthogonal Solutions of $q^{-1}$-Periodic Equations
(378K, .pdf)
ABSTRACT. The quantum calculus, otherwise known as the $q$-calculus, has been found to have a wide variety of interesting applications in computational number theory, and the theory of orthogonal polynomials, for example. As such, herein we investigate a class of entire functions that are $q^{-1}$-orthogonal with respect to their own zeros, and find that in this equivalence class, the only $q^{-1}$-periodic functions are nonzero constant-valued functions. It is well understood by the Fundamental Theorem of Algebra, that a nonzero constant function has no roots. Accordingly, this study aims to develop a novel approach to the field of $q^{-1}$-orthogonal polynomials, and the distribution of their zeros.