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Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions

Conrad, Eric van Fossen
http://rave.ohiolink.edu/etdc/view?acc_num=osu1029248229

Abstract Details

2002, Doctor of Philosophy, Ohio State University, Mathematics.
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 ontinued 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 pecializing 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.
Stephen Milne (Advisor)
Mathematics
Jacobi elliptic functions; Dixon elliptic functions; Hankel determinants; Turanian determinants; persymmetric determinants; associated continued fractions; regular C-fractions; Laplace transform

Recommended Citations

Citations

  • Conrad, E. V. F. (2002). Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1029248229

    APA Style (7th edition)

  • Conrad, Eric. Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions. 2002. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1029248229.

    MLA Style (8th edition)

  • Conrad, Eric. "Some Continued Fraction Expansions of Laplace Transforms of Elliptic Functions." Doctoral dissertation, Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1029248229

    Chicago Manual of Style (17th edition)

osu1029248229
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© 2002, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.