Special Functions and Orthogonal Expansions with Applications to Nonlinear Dynamics and Quantum Mechanics

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This project combines the efforts of two investigators working on problems involving orthogonal expansions, special functions and their applications. The program concerns investigations into qualitative and quantitative features of orthogonal polynomials in several variables and polynomials orthogonal with respect to complex weight functions. Work is also planned on the application of elliptic functions to study spectral properties of orthogonal polynomials arising from stochastic processes and perturbation theory. Additional work will be done investigating algorithms which reduce the complexity of equations describing quantum mechanics. In particular, the applicability of Pade approximants or their radial simplifications will be considered. Related work involves the distribution of roots of complex orthogonal polynomials and Pade approximants. The latter may shed light on the poles of Pade approximants in understanding resonance phenomena in non-linear dynamics. While some of this research represents continuations of earlier work, that relating to multivariate polynomials follows recent breakthroughs following a period of intense research by a number of mathematicians. Only during the past year have multivariate analogues of classical one-variable results been attained. These include multivariate integrals extending the well-known gamma and beta integrals. The complexity of this research will require advanced computational support for testing and experimentation.