Given a sequence of complex square matrices, @math , consider the sequence of their partial products, defined by @math . What can be said about the asymptotics as @math of the sequence @math , where @math is a continuous function? A special case of our most general result addresses this question under the assumption that the matrices @math are an @math perturbation of a sequence of matrices with bounded partial products. We apply our theory to investigate the asymptotics of the approximants of continued fractions. In particular, when a continued fraction is @math limit 1-periodic of elliptic or loxodromic type, we show that its sequence of approximants tends to a circle in @math , or to a finite set of points lying on a circle. Our main theorem on such continued fractions unifies the treatment of the loxodromic and elliptic cases, which are convergent and divergent, respectively. When an approximating sequence tends to a circle, we obtain statistical information about the limiting distribution of the approximants. When the circle is the real line, the points are shown to have a Cauchy distribution with parameters given in terms of modifications of the original continued fraction. As an example of the general theory, a detailed study of a @math -continued fraction in five complex variables is provided. The most general theorem in the paper holds in the context of Banach algebras. The theory is also applied to @math -matrix continued fractions and recurrence sequences of Poincar 'e type and compared with closely related literature.
We are aware of four other places where work related to the results of this section was given previously. Two of these were motivated by the identity of Ramanujan. The first paper is @cite_46 which gave the first proof of . The proof in @cite_46 is particular to the continued fraction . However, section 3 of @cite_46 studied the recurrence @math and showed that when @math , the sequence @math has six limit points and that moreover a continued fraction whose convergents satisfies this recurrence under the @math assumption tends to three limit points (Theorem 3.3 of @cite_46 ). The paper does not consider other numbers of limits, however. Moreover, the role of the sixth roots of unity in the recurrence is not revealed. In the section 6 of the present paper, we treat the general case in which recurrences can have a finite or uncountable number of limits. Previously in @cite_20 we treated such recurrences with a finite number @math of limits as well as the associated continued fractions.
@cite_46 @cite_20
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Abstract On page 45 in his lost notebook, Ramanujan asserts that a certain q -continued fraction has three limit points. More precisely, if A n B n denotes its n th partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A n B n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established. For integers m⩾2, 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 Poincare-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, k⩾2. 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 m⩾2.