Continued fractions for some alternating series

We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen's constant $$C = \sum\limits_{i \geqslant 0} {\frac{{( - 1)^i }}{{S_i - 1}}}$$ is transcendental. Here (S _n ) isSylvester's sequence defined byS ₀=2 andS _n+1 =S _n ² ?S _n +1 forn≥0. We also explicitly compute the continued fraction for the numberC; its partial quotients grow doubly exponentially and they are all squares.     

Continued fractions for certain algebraic power series

Let F be an arbitrary field and let K = F((x⁻¹)) be the field of formal Laurent series in x⁻¹ over F. The usual theory of continued fractions carries over to K, with the polynomials in x playing the role of the integers. We study the continued fraction expansions of elements of K which are algebraic over F(x), the field of rational functions of x.We give the first explicit expansions of algebraic elements of degree greater than 2 for which the degrees of the partial quotients are bounded. In particular we give explicitly the continued fraction expansion for the solution f in K of the cubic equation xf³ + f + x = 0 when F = GF(2). This cubic was studied by Baum and Sweet. We give examples, for every field F of characteristic greater than 2, of algebraic elements of degree greater than 2 whose partial quotients are all linear, and we give these expansions explicitly. These are the first known examples with partial quotients of bounded degree when F has characteristic greater than 2.     

Continued fractions and the polygamma functions

In a previous study we have shown that the polygamma functions (derivatives of the logarithm of the gamma function) relate to Stieltjes transforms in the square of the argument. These transforms in turn may be converted to Stieltjes continued fractions; in the background is a determined Stieltjes moment problem.In the present study we use the Hamburger form of the Stieltjes integral to produce a set of real monotonic increasing and monotonic decreasing approximants to each of the real and imaginary parts of a polygamma function when the argument is complex. The approximants involve rational fractions which appear to be new.Special attention is given to ln Γ(z) and the psi function.     

Continued fractions of operator-valued analytic functions

The subject of this work is the convergence of infinite continued fractions whose coefficients are analytic functions which take their values in the space of bounded linear operators on a complex Hilbert space. By appropriately combining a prior result of Fair with Vitali's theorem, we show that convergence under the uniform operator topology occurs on certain angular regions of the complex plane whenever the operator coefficients are commutative, invertible, and satisfy certain conditions on their numerical ranges.     

Continued fractions for quadratic elements in formal power series

There exist certain quadratic elements α∈?((t ^?1)) over the rational function field ?(t) having nonperiodic continued fraction expansion, see W.M. Schmidt in (Acta Arith. 95(2):139–166, 2000). Hence we need a modification of Lagrange’s theorem with regard to function fields instead of number fields. In this paper, we introduce a class of continued fractions and describe Lagrange’s theorem as a conjecture related to quadratic elements over ?(t). We give some examples which support our conjecture.     

Continued fractions for linear fractional transformations of power series

We present an algorithm to produce the continued fraction expansion of a linear fractional transformation of a power series. Giving an application, we demonstrate that the behavior of the algorithm is intimately related with the continued fraction expansions of certain algebraic power series over finite fields.     

Continued fractions for hyperquadratic power series over a finite field

An irrational power series over a finite field of characteristic p is called hyperquadratic if it satisfies an algebraic equation of the form x=(Ax^r+B)/(Cx^r+D), where r is a power of p and the coefficients belong to . These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific as in the classical case, but more sophisticated. Here we present a general result on the way some of these expansions are generated. We apply it to describe several families of expansions having a regular pattern.     

Continued fractions for some transcendental numbers

We consider series of the form

$$\begin{aligned} \frac{p}{q} +\sum _{j=2}^\infty \frac{1}{x_j}, \end{aligned}$$

where \(x_1=q\) and the integer sequence \((x_n)\) satisfies a certain non-autonomous recurrence of second order, which entails that \(x_n|x_{n+1}\) for \(n\ge 1\). It is shown that the terms of the sequence, and multiples of the ratios of successive terms, appear interlaced in the continued fraction expansion of the sum of the series, which is a transcendental number.

     

Continued fractions and irrational rotations

Let \(\alpha \in (0, 1)\) be an irrational number with continued fraction expansion \(\alpha =[0; a_1, a_2, \ldots ]\) and let \(p_n/q_n= [0; a_1, \ldots , a_n]\) be the nth convergent to \(\alpha \). We prove a formula for \(p_nq_k-q_np_k\) \((k<n)\) in terms of a Fibonacci type sequence \(Q_n\) defined in terms of the \(a_n\) and use it to provide an exact formula for \(\{n\alpha \}\) for all n.     

Continued fractions with multiple limits

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.     

Continued fractions and the Markoff tree

We give here a full account of Markoff's celebrated result on badly approximable numbers. The proofs rely exclusively on the classical theory of simple continued fractions, together with Harvey Cohn's method using words in the free group with two generators for the determination of the structure of periods of the continued fractions of Markov irrationals. Appendix A gives a short self-contained presentation of the results on continued fractions used here and Appendix B gives short proofs of some results on the still open uniqueness problem for Markoff numbers.     

Continued fractions with bounded partial quotients

This paper gives the exact bound of the continued fraction expansion of when has bounded partial quotients and is a Möbius transformation where all entries are integers.

     

Continued fractions with three limit points

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 nth 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.     

Ramanujan-type congruences for certain generating functions

For nonnegative integers a, b, the function d _a,b (n) is defined in terms of the q-series $\sum_{n=0}^\infty d_{a,b}(n)q^n=\prod_{n=1}^\infty{(1-q^{ an})^b}/{(1-q^n)}$ . We establish some Ramanujan-type congruences for d _a,b (n) by the theory of modular forms with complex multiplication. As consequences, we generalize the famous Ramanujan congruences for the partition function p(n) modulo 5, 7, and 11.