We generalise remarks of Euler and of Perron by explaining how to detail all quadratic integers for which the symmetric part of their continued fraction expansion commences with prescribed partial quotients. I last saw Bela Brindza, my once postdoctoral student, in April, 2002. I was working on the paper below and attempted to enthuse him with its results, particularly those concerning periodic...

We investigate the relationships among the Pad e table, continued fraction expansions and perfect reconstruction (PR) lter banks. We show how the Pad e table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) lter banks. This is achieved through characterization of all two-channel bi-orthogonal PR lter banks. The parameterization...

The Gauss 1812 problem gave rise to an extended literature. In modern times, the socalled Gauss-Kuzmin-Lévy theorem is still one of the most important results in the metrical theory of regular continued fractions (RCFs). A recent survey of this topic is to be found in [10]. From the time of Gauss, a great number of such theorems followed. See, for example, [2, 6, 7, 8, 18]. Apart from the RCF e...

In some recent papers, the authors considered regular continued fractions of the form [a0; a, · · · , a } {{ } m , a, · · · , a } {{ } m , a, · · · , a } {{ } m , · · · ], where a0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived from ...

The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have derived a space-time generalized telegrapher’s equation with a fractional nested hiera...

We display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series ∑ chX −h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple arguments, let alone examples, demonstrate...

We prove a continued fraction expansion for a certain q–tangent function that was conjectured by Prodinger.

(a general reference is Chapter I of [9]). If ξ is irrational, then, by letting X tend to infinity, this provides infinitely many rational numbers x1/x0 with |ξ − x1/x0| ≤ x 0 . By contrast, an irrational real number ξ is said to be badly approximable if there exists a constant c1 > 0 such that |ξ − p/q| > c1q for each p/q ∈ Q or, equivalently, if ξ has bounded partial quotients in its continue...

We note that the continued fraction expansion of a lacunary formal power series is a folded continued fraction with monomial partial quotients, and with the property that its convergents have denominators that are the sums of distinct monomials, that is, they are polynomials with coefficients 0, 1, and −1 only. Our results generalise, simplify and refine remarks of a previous note ‘Convergents ...