In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.

Let $K=\mathbb{Q}(\sqrt{m})$ be an imaginary quadratic field with $O_K$ its ring of integers. Let $\pi$ and $\beta$ be an irreducible element and a nonzero element, respectively, in $O_K$. In the authors' earlier work, it was proved for the cases, $m\not\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ and $m\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ that if $\pi ={\alpha }_n{\beta }^n+{\alpha }_{n-1}{\beta }^{n-1}+\cdots +{\alpha }_1\beta +{\alpha }_0=:f(\beta )$, where $n\ge 1$, ${\alpha }_n\in O_K\setminus \{0\}$, ${\alpha }_0,\dots ,{\alpha }_{n-1}$ belong to a complete residue system modulo $\beta$, and the digits ${\alpha }_{n-1}$ and ${\alpha }_n$ satisfy certain restrictions, then the polynomial $f(x)$ is irreducible in $O_K[x]$. In this paper, we extend these results by establishing further irreducibility criteria for polynomials in $O_K[x]$. In addition, we provide elements of $\beta$ that can be applied to the new criteria but not to the previous ones.