Is there a term for continued fractions of the form $[b; b, b, b \dots ]$?

Question

We denote the continued fraction $$a_0 + \frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}$$ in shorthand as $[a_0;a_1,a_2 \dots]$ or as $[a_0;a_1,a_2 \dots a_n]$ if it is finitely nested. I would like to know if there is a term for continued fractions of the form $[a;a,a\dots]$ and/or $[0;a,a\dots]$ where the constants are all equal (either finitely or infinitely nested.)

Answer 1

In the infinite case, we have $$[a;a,a,...] = a + \frac{1}{[a;a,a,...]} $$ and so $$[a;a,a,...]^2 - a [a;a,a,...] - 1 = 0 $$ Applying the quadratic formula we get $$[a;a,a,...] = \frac{a + \sqrt{a^2+4}}{2} $$ I suppose that's a shorthand for $[a;a,a,...]$ in some sense.

Answer 2

A periodic continued fraction (with period $m$) is a continued fraction of the form $$\left[a_0; a_1, \dotsc, a_{n}, \overline{a_{n+1}, \dotsc, a_m}\right],$$ i.e. it is a continued fraction in which the same $m$ terms repeat indefinitely after some initial block of terms. A periodic continued fraction with no initial block, i.e. an $m$-periodic continued faction of the form $$ \left[ \overline{a_0; a_1, \dotsc, a_{m-1}} \right] $$ is purely periodic continued fraction.^[1] Therefore a continued fraction of the form $$ \left[ a_0; \overline{a_0} \right] $$ is a purely periodic continued fraction with period $1$.

I am not aware of any specific terminology for terminating continued fractions in which a single term is repeated—it might be reasonable to call such a thing a purely periodic continued fraction of length $n$, though "periodic" tends to imply that the terms repeat infinitely often, so one would need to be very clear about the meaning of the terminology.

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[1] I cannot find an authoritative reference for this terminology, but a .pdf from an MIT OpenCourseware class on number theory uses this terminology (the first occurrence is at the bottom of page 3).