A generalization of continued fractions

Continued fraction expansions with variable numerators

“…2 continued fractions of the form (1) corresponding to a sequence (a n ) n≥1 which is non-proper; i.e., for which we have that b n < a n for at least one n ≥ 1. As in [1] we show that if we drop the demand that the expansion is proper, and if a n ≥ 2 infinitely often, every x between 0 and a 1 has infinitely many expansions of the form (1), of which at least one is infinite. Furthermore, if infinitely often we have that a n ≥ 3 we have that every x between 0 and a 1 has infinitely many infinite expansions of the form (1).…”

“…In case x is irrational this obviously yields an infinite sequence (b n ) n≥0 for which b n ≥ a n for n ≥ 1 and T a n T a n−1 · · · T a 1 (x − b 0 ) · · · = a n T a n−1 T a n−2 · · · T a 1 …”

“…(1) Note that the case a n = 1 (for n ≥ 1) is the RCF case. In case x is rational, Leighton's algorithm yields that the continued fraction expansion (1) of x is finite, while it is infinite when x is irrational.…”

“…However, in their result, the continued fraction expansion (which is now a so-called "N -expansion") is not necessarily proper, i.e., we need not have that b n ≥ N for all n ≥ 1. In fact, Anselm and Weintraub showed in [1] that if we drop the demand that the expansion is proper, every x between 0 and N has infinitely many N -expansions, i.e., expansions of the form…”

“…Proper N -expansions are called best expansions by Anselm and Weintraub [1]. In [9] it is shown that these best expansions for (fixed) N ∈ N can be obtained via the transformation T N : [0, N ] → [0, N ], defined by…”