Transcendence of Jacobi's theta series

On the complexity of algebraic numbers I. Expansions in integer bases

“…Our method does not apply to show the transcendence of α. Let us mention that this α is known to be transcendental: this is a consequence of deep transcendence results proved in [10] and in [17], concerning the values of theta series at algebraic points. The proofs of Theorems 1 to 4 are given in Section 5.…”

Again on the Irrationality of a Certain Infinite Product

“…oo to show that E q (-1) = JJ (1 -q~]) is not algebraic of degree less than 3 for any q € 2Z ¿=i with |g| > 1. In the same year 1996 there appeared Nesterenko's [7], [8] deep transcendence theorems on modular functions from which one can quite easily deduce (much more than) the transcendency of E q (-1) for any algebraic number q outside the unit circle, for details see Duverney, the two Nishioka's, and Shiokawa [4], To conclude these few remarks on the present state of the arithmetical studies on E q (x), we point out that Nesterenko's analytic machinery works for some (simple) fixed χ with q as the variable, whereas in every work in this topic before 1996 the parameter q is fixed and χ is the complex variable.…”

Algebraic relations for reciprocal sums of Fibonacci numbers

“…Nishioka, Ku. Nishioka, and the last named author [9] (see also [8]) proved the transcendence of the numbers by using Nesterenko's theorem on the Ramanujan functions P (q), Q(q), and R(q) (see Section 3).…”