Jun-ichi Tamura

Arithmetical properties of a certain power series

Abstract The function f(θ, φ; x, y) = Σk = 1∞ Σ1 ≤ m ≤ kθ + φ xkym, where θ > 0 is irrational and φ is real, satisfies Mahler-type functional equations which enable us to represent it by a gap-like series and then by a continued fraction. Using these representations, we describe the sequence {[(k + 1) θ + φ] − [kθ + φ]}k = 1∞ by a chain of substitutions and give algebraic independence results for the values of f(θ, φ, x, y) at some algebraic points when the partial quotients of the continued fraction of θ are unbounded, and irrationality measures for the values at some rational points.

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.