S. Chakraborty and B.V. Rao have introduced the continued fraction expansion of a number in terms of an irrational \(\theta \in (0, 1)\) . This new expansion of positive reals, different from the regular continued fraction expansion, is called \(\theta \) -continued fraction expansion or \(\theta \) -expansion for short. We mention that the case \(\theta = 1\) refers to regular continued fraction (RCF) expansions.