We study the question: How often do partial sums of power series of functions coalesce with convergents of the (simple) continued fractions of the functions? Our theorems quantitatively demonstrate that the answer is: not very often. We conjecture that in most cases there are only a finite number of partial sums coinciding with convergents. In many of these cases, we offer exact numbers in our conjectures.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-3-4,
author = {Bruce C. Berndt and Sun Kim and M. Tip Phaovibul and Alexandru Zaharescu},
title = {Diophantine approximation with partial sums of power series},
journal = {Acta Arithmetica},
volume = {161},
year = {2013},
pages = {249-266},
zbl = {1318.11088},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-3-4}
}
Bruce C. Berndt; Sun Kim; M. Tip Phaovibul; Alexandru Zaharescu. Diophantine approximation with partial sums of power series. Acta Arithmetica, Tome 161 (2013) pp. 249-266. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa161-3-4/