Symbolic computation has been widely applied to Combinatorics, Number Theory, and also other fields. Many reliable and fast algorithms with corresponding implementations now have been established and developed. Using the tool of Experimental Mathematics, especially with the help of mathematical software, in particularly Mathematica, we could visualize the data, manipulate algorithms and implementations. The work presented here, based on symbolic computation, involves the following two parts. The first part introduces a systematic integration method, called the Method of Brackets. It only consists of a small number of simple and direct rules coming from the Schwinger parametrization of Feynman diagrams. Verification of each rule makes this method rigorous. Then it follows a necessary theorem that different series representations of the integrand, though lead to different processes of computations, do not affect the result. Examples of application lead to further discussions on analytic continuation, especially on Pochhammer symbol, divergent series and connection to Mellin transform of the Method of Brackets. In the end, comparison with other integration methods and a Mathematica package manual are presented. The second part provides a symbolic approach on the study of Bernoulli numbers and its generalizations. The Bernoulli symbol $\mathcal{B}$ originally comes from Umbral Calculus, as a formal approach to Sheffer sequences. Recently, a rigorous footing by probabilistic proof makes it also a random variable with its density function a shifted hyperbolic trigonometric function. Such an approach together with general method on random variables gives a variety of results on generalized Bernoulli polynomials, multiple zeta functions, and also other related topics. / Lin Jiu
| Identifer | oai:union.ndltd.org:TULANE/oai:http://digitallibrary.tulane.edu/:tulane_49506 |
| Date | January 2016 |
| Contributors | Jiu, Lin (author), Moll, Victor (Thesis advisor), School of Science & Engineering Mathematics (Degree granting institution) |
| Publisher | Tulane University |
| Source Sets | Tulane University |
| Language | English |
| Detected Language | English |
| Type | Text |
| Format | electronic |
| Rights | Embargo |
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