It is well known that the continued fraction expansion of $\sqrt{D}$ readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of $x^2-D{y}^2=\pm 1$. Here we notice that, analogously, the point halfway to a solution of $x^2-D{y}^2=-3$ can be recognised. We explain what is going on.

Many new types of Hurwitz continued fractions have been studied by the author. In this paper we show that all of these closed forms can be expressed by using confluent hypergeometric functions ${}_0{F}_1(;c;z)$. In the application we study some new Hurwitz continued fractions whose closed form can be expressed by using confluent hypergeometric functions.