SEL Egyptian fraction expansion and characterizations of rational numbers

Abstract

Abstract In the earlier work, the SEL series expansion for any real number is constructed and characterizations of rational numbers by using such expansion are established. In this paper, we construct the SEL Egyptian fraction expansion for any real number and establish characterizations of rational numbers by using such expansion. These results yield a generalized version of the results for the Fibonacci-Sylvester expansion and the Cohen-Egyptian fraction expansion. A new expansion, called the Lüroth Egyptian fraction expansion is obtained and the relation between the SEL Egyptian fraction expansion and the SEL series expansion is also established.