On the harmonic continued fractions (original) (raw)

Closed form expressions for two harmonic continued fractions

The Mathematical Gazette, 2017

A continued fraction is an expression of the formThe expression can continue for ever, in which case it is called an infinite continued fraction, or it can stop after some term, when we call it a finite continued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

On certain continued fractions involving two basic hypergeometric series

The object of this paper is to establish continued fraction for the ratio of two 3 4  series by the application of some known transformation formula in basic hyper geometric series. Keywords and phrases: Basic hyper geometric series, Continued fractions and some known results in basic hyper geometric series.

On a continued fraction expansion for Eulerʼs constant

Journal of Number Theory, 2013

Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant γ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's approximations in terms of Meijer G-functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and γ with rational coefficients. Using this construction we find new rational approximations to γ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Euler's constant. It seems to be the first non-trivial continued fraction expansion convergent to Euler's constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler-Gompertz constant found by Stieltjes in 1895. m n+m m and satisfy the second-order linear recurrence relation (n + 1) 3 y n+1 − (34n 3 + 51n 2 + 27n + 5)y n + n 3 y n−1 = 0

Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan's Identity

2016

We consider integer sequences connected to the famous Laplace continued fraction for the function R(t) = ∫∞ t φ(x)dx/φ(t), where φ(t) = e −t2/2/ √ 2π is the standard normal density. We compute the generating functions for these sequences and study their relation to the Hermite and Bessel polynomials. Using the master equation for the generating functions, we find a new proof of the Ramanujan identity.

Continued fractions with multiple limits

Advances in Mathematics, 2007

For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem.

A simple proof of a remarkable continued fraction identity

Proceedings of the American Mathematical Society, 1995

We give a simple proof of a generalization of the equality Y J-= [0, 2°, 2l, 2l, 22 , 2\ 25, ...], 2l"/rl where t = (1 + \/5)/2 and the exponents of the partial quotients are the Fibonacci numbers, and some closely related results.