For given nonsquare positive integers C ≡ 5(mod 8), we investigate families {Dk(X)}k∈N of integral polynomials of the form Dk(X) = AkX +2BkX+C where (Bk/2)−(Ak/2)C = 4, and show that the period length of the simple continued fraction expansion of (1+ √ Dk(X))/2 is a multiple of k, and independent ofX. For each member of the families involved, we show how to easily determine the fundamental unit...

The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an m-continued fraction converge to a multiformal Laurent series, and are best rational approximations to it; conversely for any multi-formal Laurent series an algorith...

We examine various properties of the continued fraction expansions of matrix eigenvector slopes of matrices from the SL(2, Z) group. We calculate the average period length, maximum period length, average period sum, maximum period sum and the distributions of 1s 2s and 3s in the periods versus the radius of the Ball within which the matrices are located. We also prove that the periods of contin...

We study the combinatorics of a continued fraction formula due to Wall. We also derive the orthogonality of little q-Jacobi polynomials from this formula, as Wall did for little q-Laguerre polynomials.

A classical result of Khinchin says that for almost all real numbers α, the geometric mean of the first n digits ai(α) in the continued fraction expansion of α converges to a number K ≈ 2.6854520 . . . (Khinchin’s constant) as n → ∞. On the other hand, for almost all α, the arithmetic mean of the first n continued fraction digits ai(α) approaches infinity as n → ∞. There is a sequence of refine...

The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients {an} and the continued-fraction coefficients {bn}. In many instances it turns out that this nonlinear relation transforms a complicated sequence {an} into a very simple one {bn}. We illustrate this simplification in the context of graph combinatorics. PACS numbers: 02.90...

We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and lower fluctuation process. Also a large deviation asymptotic for single digits is given.

The derivative of a finite continued fraction of a complex variable is derived by presenting the continued fraction as a component of a finite composition of Ĉ → Ĉ linear fractional transformations of analytic functions. Connections to previous work and possible applications of the deduced formula are briefly discussed.

It has been believed that the continued fraction expansion of (α, β) (1, α, β is a Q-basis of a real cubic field) obtained by the modified JacobiPerron algorithm is periodic. We conducted a numerical experiment (cf. Table B, Figure 1 and Figure 2) from which we conjecture the non-periodicity of the expansion of (⟨ 3 √3⟩, ⟨ 3 √9⟩) (⟨x⟩ denoting the fractional part of x). We present a new algorit...

On page 26 in his lost notebook, Ramanujan states an asymptotic formula for the generalized Rogers–Ramanujan continued fraction. This formula is proved and made slightly more precise. A second primary goal is to prove another continued fraction representation for the Rogers–Ramanujan continued fraction conjectured by R. Blecksmith and J. Brillhart. Two further entries in the lost notebook are e...