Tasoev's continued fractions and Rogers-Ramanujan continued fractions

We show some new variations on Tasoev's continued fractions , where the periodic parts include the exponentials in k instead of the polynomials in k. We also mention some relations with other kinds of continued fractions, in particular, with Rogers-Ramanujan continued fractions.     

Quasi-periodic continued fractions

Using recent work of Adamczewski and Bugeaud, we are able to relax the conditions given by Baker to establish transcendence in the class of quasi-periodic continued fractions.     

Matrix-valued continued fractions

We discuss the properties of matrix-valued continued fractions based on Samelson inverse. We begin to establish a recurrence relation for the approximants of matrix-valued continued fractions. Using this recurrence relation, we obtain a formula for the difference between mth and nth approximants of matrix-valued continued fractions. Based on this formula, we give some necessary and sufficient conditions for the convergence of matrix-valued continued fractions, and at the same time, we give the estimate of the rate of convergence. This paper shows that some famous results in the scalar case can be generalized to the matrix case, even some of them are exact generalizations of the scalar results.     

Transcendental continued fractions

In two previous papers Nettler proved the transcendence of the continued fractions $\text{A := a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+ ?}$, $\text{B := b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+ ?}$ as well as the transcendence of the numbers A + B, A ? B, AB, $\text{A}\text{B}$ where the a's and b's are positive integers satisfying a certain mutual growth condition. In the present paper even the algebraic independence of A and B is proved under almost the same condition and furthermore a result concerning the transcendency of A^B is established.     

Transcendental continued fractions

In a previous paper it was proven that given the continued fractions

$\text{A = a}{}_1\text{+}\text{1}\text{a}{}_2\text{+}\text{1}\text{a}{}_3\text{+… and B = b}{}_1\text{+}\text{1}\text{b}{}_2\text{+}\text{1}\text{b}{}_3\text{+…}$

where the a's and b's are positive integers, then A, B, A ± B, $\text{A}\text{B}$ and AB are irrational numbers if $\text{a}{}_{\mathrm{n}}\text{2}\text{> b}{}_{\mathrm{n}}\text{> a}{}_{\mathrm{n?1}}{}^{\mathrm{5n}}$ for all n sufficiently large, and transcendental numbers if for all n sufficiently large. Using a more direct approach it is proven in this paper that A, B, A ± B, $\text{A}\text{B}$ and AB are transcendental numbers if a_n > b_n > a_n?1^(n?1)2 for all n sufficiently large.     

Relation between interpolating integral continued fractions and interpolating branched continued fractions

We construct and investigate an interpolating integral continued fraction, which represents a natural generalization of interpolating continued fractions. We also point out the optimal choice of the sequence of interpolating knots.     

A family of continued fractions

We investigate a one-parameter family of infinite generalised continued fractions. The fractions converge to rational values which can be explicitly evaluated. The sequences of numerators and denominators are already to be found in the Online Encyclopedia of Integer Sequences with reference to various topics - but not to continued fractions.     

Convergence of matrix continued fractions

The aim of this work is to give some criteria on the convergence of matrix continued fractions. We begin by presenting some new results which generalize the links between the convergent elements of real continued fractions. Secondly, we give necessary and sufficient conditions for the convergence of continued fractions of matrix arguments. This paper will be completed by illustrating the theoretical results with some examples.     

A generalization of continued fractions

We investigate a generalization of classical continued fractions, where the “numerator” 1 is replaced by an arbitrary positive integer N. We find both similarities to and surprising differences from the classical case.     

Algebraic interpretation of continued fractions

An alternative (equivalent) definition of continued fractions in terms of a group representation is introduced. With this definition, continued fractions are considered as sequences in a topological group, converging (in some sense) to its boundary. This point of view yields an alternative (equivalent) proof for Lane's convergence theorem for periodic continued fractions.     

On the harmonic continued fractions

The Ramanujan Journal - In this paper, we study the harmonic continued fractions. These form an infinite family of ordinary continued fractions with coefficients $$\frac{t}{1}, \frac{t}{2},...     

Snake graphs and continued fractions

This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the continued fractions as well as the Euclidean division algorithm. We apply our findings to establish results on sums of squares, palindromic continued fractions, Markov numbers and other statements in elementary number theory.     

Hypergeometric series and continued fractions

Ramanujan’s results on continued fractions are simple consequences of three-term relations between hypergeometric series. Theirq-analogues lead to many of the continued fractions given in the ‘Lost’ notebook in particular the famous one considered by Andrews and others.     

Some more predictable continued fractions

The continued fraction expansions of a certain class of numbers of the form. $$u^{ - k_0 } \pm u^{ - k_1 } \pm u^{ - k_2 } \pm ...$$ have a definite pattern.