Continued fractions and modular functions
نویسندگان
چکیده
منابع مشابه
Continued Fractions and Modular Functions
It is widely recognized that the work of Ramanujan deeply influenced the direction of modern number theory. This influence resonates clearly in the “Ramanujan conjectures.” Here I will explore another part of his work whose position within number theory seems to be less well understood, even though it is more elementary, namely that related to continued fractions. I will concentrate on the spec...
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This incursion into the realm of elementary and probabilistic number theory of continued fractions, via modular forms, allows us to study the alternating sum of coeecients of a continued fraction, thus solving the longstanding open problem of their limit law.
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Found in the collected works of Eisenstein are twenty continued fraction expansions. The expansions have since emerged in the literature in various forms, although a complete historical account and self-contained treatment has not been given. We provide one here, motivated by the fact that these expansions give continued fraction expansions for modular forms. Eisenstein himself did not record p...
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This paper extends Flajolet’s (Discrete Math. 32 (1980) 125-161) combinatorial theory of continued fractions by obtaining the generating function for paths between horizontal lines, with arbitrary starting and ending points and weights on the steps. Consequences of the combinatorial arguments used to determine this result are combinatorial proofs for many classical identities involving continue...
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This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstraß function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordin...
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 2005
ISSN: 0273-0979
DOI: 10.1090/s0273-0979-05-01047-5