Symmetric continued fractions related to certain series

Abstract

Let f(x) ∈ Z [ x] with positive leading coefficient and of degree ≥ 2. We define f m ( x) to be the m-fold iteration of f( x); i.e., f 0( x) ≡ x and f m ( x) = f( f m − 1 ( x)), m ∈ N . Denote by θ n ( x; f) the formal Cantor series θ n(x;f)= ∑ m−0 n 1 f 0(x)f 1(x)…f m(x) and define θ( x; f) = lim n → ∞ θ n ( x; f) for those x ∈ N for which the limit exists. 1. (i) If f( x) is of the form x( x + 2)( x − 2) g( x) + x 2 − 2 with g(x) ∈ Z [ x], we can determine the simple continued fraction expansions of θ n ( x; f) and of θ( x; f) for any integer x ≥ x 1( f), where x 1( f) is the least integer ≥ 3 with f( x) > 2 x − 2 for all x ≥ x 1( f). We also show that θ( x; f) is transcendental for any such x except when g( x) ≡ 0 in which case θ( x; x 2 − 2) is a quadratic irrational for all x ≥ 3. 2. (ii) If f( x) is also of the form x 2( x + 2)( x − 2) g( x) + x 2 − 2 with g(x) ∈ Z [ x], g( x) n= 0, we can, in addition, determine the simple continued fraction expansions of θ n(x; f) x and of θ(x; f) x for any integer x ≥ x 2( f), where x 2( f) is the least integer ≥ 3 with f( x) > 2 x 2 − 2 for all x ≥ x 2( f). For the cases n = 1, 2, and 3, we also prove that f( x) is necessarily of the form (i) for the continued fraction expansion of θ n ( x; f) ( x ≥ x 1, x ∈ N ) to be symmetric and that f( x) is necessarily of the form (ii) for the continued fraction expansion of θ n(x; f) x ( x ≥ x 2, x ∈ N ) to be symmetric.