Mathematical Proceedings of the Cambridge Philosophical Society



On Tasoev's continued fractions


TAKAO KOMATSU a1
a1 Faculty of Education, Mie University, Tsu, Mie 514-8507, Japan. e-mail: komatsu@edu.mie-u.ac.jp

Abstract

Let $a$ be a positive integer with $a > 1$. We show \[ [0;a,a^2,a^3,a^4,\ldots]=\frac{\sum\nolimits^{\infty}_{s=0}a^{-(s+1)^2}\prod\nolimits^s_{i=1}(a^{2i}-1)^{-1}}{\sum\nolimits^{\infty}_{s=0}a^{-s^2}\prod\nolimits^s_{i=1}(a^{2i}-1)^{-1}} \] and \[ [0;a,a,a^2,a^2,a^3,a^3,\ldots]=\frac{\sum\nolimits^{\infty}_{s=0}a^{-\frac{(s+1)(s+2)}{2}}\prod\nolimits^s_{i=1}(a^{i}-1)^{-1}}{\sum\nolimits^{\infty}_{s=0}a^{-\frac{s(s+1)}{2}}\prod\nolimits^s_{i=1}(a^{i}-1)^{-1}} \] A more general case is discussed.

(Received April 9 2001)
(Revised June 26 2001)