Applying Zeilberger’s algorithm of creative telescoping to a family of certain very-well-poised hypergeometric series involving linear forms in Catalan’s constant with rational coefficients, we obtain a second-order difference equation for these forms and their coefficients. As a consequence we derive a new way of fast calculation of Catalan’s constant as well as a new continued-fraction expans...

Continued fractions lie at the heart of a number of classical algorithms like Euclid's greatest common divisor algorithm or the lattice reduction algorithm of Gauss that constitutes a 2-dimensional generalization. This paper surveys the main properties of functional operators, |transfer operators| due to Ruelle and Mayer (also following L evy, Kuzmin, Wirsing, Hensley, and others) that describe...

– Let ξ be a real number and let n be a positive integer. We define four exponents of Diophantine approximation, which complement the exponents w n (ξ) and w * n (ξ) defined by Mahler and Koksma. We calculate their six values when n = 2 and ξ is a real number whose continued fraction expansion coincides with some Sturmian sequence of positive integers, up to the initial terms. In particular, we...

Dumont and Foata introduced in 1976 a three-variable symmetric refinement of Genocchi numbers, which satisfies a simple recurrence relation. A six-variable generalization with many similar properties was later considered by Dumont. They generalize a lot of known integer sequences, and their ordinary generating function can be expanded as a Jacobi continued fraction. We give here a new combinato...

Solving Pell’s equation is of relevance in finding fundamental units in real quadratic fields and for this reason polynomial solutions are of interest in that they can supply the fundamental units in infinite families of such fields. In this paper an algorithm is described which allows one to construct, for each positive integer n, a finite collection, {Fi}, of multi-variable polynomials (with ...

Let z be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums S(pk, qk) belonging to convergents pk/qk of the regular continued fraction expansion of z with that of Dedekind sums S(sj/tj) belonging to convergents sj/tj of the negative regular continued fraction expansion of z. Whereas the three main cases of this behavior are closely related, a more detailed study of...

The statistics of the digits of a continued fraction, also known as partial quotients, have been studied at least since the time of Gauss. The usual measure m on the open interval (0, 1) gives a probability space U . Let ak, k ≥ 1 be integer-valued random variables which take α ∈ (0, 1) to the k partial quotient or digit in the continued fraction expansion α = 1/(a1 + 1/(a2 + · · · )). Let Sr =...

We provide a systematic procedure for generating the coefficients of continued fraction expansion test function associated with characteristic polynomial stable system difference equations. illustrate feasibility procedure, and we an application on stability two-dimensional digital filters.

The evenness and the values modulo 4 of lengths periods continued fraction expansions p 2p for p≡3(mod4) a prime are known. Here we prove similar results expansion pq, where p,q≡3(mod4) distinct primes.

A three-dimensional analogue of the connection between exponent irrationality a real number and growth partial quotients its expansion in simple continued fraction is investigated. As multidimensional generalization fractions, Klein polyhedra are considered. Bibliography: 12 titles.