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

这项工作将研究形式级数（本影微积分），目的是将结果应用于特殊函数，正交多项式和数值分析中的相关问题。该研究介于特殊函数、数值分析和组合学之间。将使用形式级数来明确地构造满足更新方程的序列的极限。当方程的右侧本身由形式级数给出时，也将考虑用形式级数显式地求解常系数非齐次差分方程。将开发此类方程的正式切比雪夫多项式解。差分方程是科学和工程中模拟重要物理过程的偏微分方程数值解的关键工具。

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.