Continued fractions with multiple limits |
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Authors: | Douglas Bowman J Mc Laughlin |
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Institution: | aNorthern Illinois University, Mathematical Sciences, DeKalb, IL 60115-2888, USA;bMathematics Department, Trinity College, 300 Summit Street, Hartford, CT 06106-3100, USA |
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Abstract: | For integers m2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem.We give a theorem on a class of Poincaré-type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity.We also generalize a curious q-continued fraction of Ramanujan's with three limits to a continued fraction with k distinct limit points, k2. The k limits are evaluated in terms of ratios of certain q-series.Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m2. |
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Keywords: | Limit periodic continued fractions q-Continued fractions Poincaré -type recurrences q-Series |
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