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.