Matrix continued fraction and expansions of the Gauss hypergeometric function

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...

New series expansions of the Gauss hypergeometric function

The Gauss hypergeometric function 2F1(a, b, c; z) can be computed by using the power series in powers of z, z/(z − 1), 1 − z, 1/z, 1/(1 − z), (z − 1)/z. With these expansions 2F1(a, b, c; z) is not completely computable for all complex values of z. As pointed out in Gil, et al. [2007, §2.3], the points z = e±iπ/3 are always excluded from the domains of convergence of these expansions. Bühring [...

A matrix Euclidean algorithm and matrix continued fraction expansions

Continued-fraction Expansions for the Riemann Zeta Function and Polylogarithms

It appears that the only known representations for the Riemann zeta function ζ(z) in terms of continued fractions are those for z = 2 and 3. Here we give a rapidly converging continued-fraction expansion of ζ(n) for any integer n ≥ 2. This is a special case of a more general expansion which we have derived for the polylogarithms of order n, n ≥ 1, by using the classical Stieltjes technique. Our...

On the real quadratic fields with certain continued fraction expansions and fundamental units

The purpose of this paper is to investigate the real quadratic number fields $Q(sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element  where $dequiv 2,3( mod  4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit$$epsilon _{d}=left(t_d+u_dsqrt{d}right) 2left.right > 1$$and  $n_d$ and $m_d...