Mathematical Sciences: Aspects of Formal Laurent Series and Applications To Some Problems in Applied Analysis

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This work will study formal series (the umbral calculus) with the aim of applying the results to special functions, orthogonal polynomials, and related problems in numerical analysis. The research lies at the interface between special functions, numerical analysis, and combinatorics. Formal series will be used to construct explicitly the limits of sequences satisfying renewal equations. The use of formal series to explicitly solve non-homogeneous difference equations with constant coefficients when the right hand side is itself given by a formal series will also be considered. Formal Chebyshev polynomial solutions for such equations will be developed. Difference equations are a key tool in the numerical solution of partial differential equations which model important physical processes in science and engineering.