A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm

In this paper, we give transcendental numbers φ and ψ such that (i) both φ and ψ have explicit g-adic expansions, and simultaneously, (ii) the vector ${\displaystyle ^t(\phi ,\psi )}$ has an explicit expression in the Jacobi-Perron algorithm (cf. Theorem 1). Our results can be regarded as a higher-dimensional version of some of the results in [1]-[5] (see also [6]-[8], [10], [11]). The numbers φ and ψ have some connection with algebraic numbers with minimal polynomials x³ - kx² - lx - 1 satisfying (1.1) k ≥ l ≥0, k + l ≥ 2 (k,l ∈ ℤ). In the special case k = l = 1, our Theorems 1-3 have been shown in [15] by a different method using the theory of representation of numbers by Fibonacci numbers of third degree.