Continued Fractions with Three Limit Points
نویسندگان
چکیده
The research described in this paper was motivated by an enigmatic entry in Ramanujan’s lost notebook [11, p. 45] in which he claimed, in an unorthodox fashion, that a certain q-continued fraction possesses three limit points. More precisely, he claimed that as n tends to ∞ in the three residue classes modulo 3, the nth partial quotients tend, respectively, to three distinct limits, which he explicitly gives. We think that there is no other example of this kind in the literature, and so we investigated the possibility of further analytic continued fractions having three distinct limit points. The purpose of this paper is to prove Ramanujan’s elusive entry, to prove a general theorem giving a class of continued fractions with three limit points, and to explicitly give further examples. To relate Ramanujan’s entry, we first introduce the customary notation
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