| In a 1907 paper, L. Rogers used two methods to obtain continued fractions for certain Laplace transforms of Jacobi elliptic functions. His first method employed repeated integration by parts, while his second method recalled an 1889 technique of T. Stieltjes. In 1996, S. Milne used these expansions and others obtained by modular transformations to derive results about sums of squares and triangular numbers.; Working independently in the 1820's, C. Jacobi and N. Abel both introduced elliptic functions to advance the study of elliptic integrals. In 1981, D. Dumont introduced symmetric variants of the elliptic functions of Jacobi and Abel to facilitate the study of certain combinatorial problems related to coefficients in Maclaurin expansions of Jacobi elliptic functions.; In this thesis, we use Dumont's elliptic functions to rederive the continued fraction expansions of Rogers. In the classical approach used by Rogers and Milne, four families of continued fractions are obtained. In our approach, members of the same four families are derived directly by specializing parameters instead of employing modular transformations.; To these four families, we add a new set of continued fractions based on certain elliptic functions that were studied in an 1890 paper by A. Dixon. These new continued fractions were discovered in 1999 in the course of work with D. Dumont. |
