Approximation of exponential function of a matrix by continued fraction expansion

A CONTINUED FRACTION EXPANSION FOR A q-TANGENT FUNCTION

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

Continued Fraction Expansions of Matrix Eigenvectors

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...

A Short Proof of the Simple Continued Fraction Expansion of

One of the most interesting proofs is due to Hermite; it arose as a byproduct of his proof of the transcendence of e in [5]. (See [6] for an exposition by Olds.) The purpose of this note is to present an especially short and direct variant of Hermite’s proof and to explain some of the motivation behind it. Consider any continued fraction [a0, a1, a2, . . .]. Its ith convergent is defined to be ...

On the Continued Fraction Expansion of a Class of Numbers

(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...

A Proof of the Continued Fraction Expansion of 1/M