#### Document Type

Article

#### Publication Date

2004

#### Abstract

This paper is an intensive study of the convergence of the Rogers-Ramanujan continued fraction. Let the continued fraction expansion of any irrational number t ∈ (0, 1) be denoted by [0, a1(t), a2(t), · · · ] and let the i-th convergent of this continued fraction expansion be denoted by ci(t)/di(t). Let S = {t ∈ (0, 1) : ai+1(t) ≥ φ di(t) infinitely often}, where φ = (√ 5 + 1)/2. Let YS = {exp(2πit) : t ∈ S}. It is shown that if y ∈ YS then the Rogers-Ramanujan continued fraction, R(y), diverges at y. S is an uncountable set of measure zero. It is also shown that there is an uncountable set of points, G ⊂ YS, such that if y ∈ G, then R(y) does not converge generally. It is further shown that R(y) does not converge generally for |y| > 1. However we show that R(y) does converge generally if y is a primitive 5m-th root of unity, some m ∈ N so that using a theorem of I. Schur, it converges generally at all roots of unity.

#### Publication Title

Transactions of the American Mathematical Society

#### ISSN

1088-6850

#### Publisher

American Mathematical Society

#### Volume

356

#### Issue

2

#### First Page

3325

#### Last Page

3347

#### Recommended Citation

Bowman, D.,
&
McLaughlin, J.
(2004).
On the Divergence of the Rogers-Ramanujan Continued Fraction on the Unit Circle.
*Transactions of the American Mathematical Society, 356*(2), 3325-3347.
Retrieved from https://digitalcommons.wcupa.edu/math_facpub/45

## Comments

Preprint version is available here.