Transcendental Numbers – Is 0.1010010001000010000010000001… Transcendental?

Question

Does anyone know if this number is algebraic or transcendental, and why?

$$\sum\limits_{n = 1}^\infty {10}^{ – n(n + 1)/2} = 0.1010010001000010000010000001 \ldots $$

Answer 1

The number $0.1010010001000010000010000001\ldots$ is transcendental.

Consider following three Jacobi theta series defined by $$\begin{align} \theta_2(q) &= 2q^{1/4}\sum_{n\ge 0} q^{n(n+1)} = 2q^{1/4}\prod_{n=1}^\infty (1-q^{4n})(1 + q^{2n})\\ \theta_3(q) &= \sum_{n\in\mathbb{Z}} q^{n^2} = \prod_{n=1}^\infty (1-q^{2n})(1+ q^{2n-1})^2\\ \theta_4(q) &= \theta_3(-q) = \sum_{n\in\mathbb{Z}} (-1)^n q^{n^2} = \prod_{n=1}^\infty (1-q^{2n})(1- q^{2n-1})^2\\ \end{align} $$ and for any $m \in \mathbb{Z}_{+}$, $k \in \{ 2, 3, 4 \}$, use $\displaystyle D^m\theta_k(q)$ as a shorthand for $\displaystyle \left( q\frac{d}{dq} \right)^m \theta_k(q)$.

Based on Corollary 52 of a survey article Elliptic functions and Transcendence by M. Waldschmidt in 2006,

Let $i, j$ and $k \in \{ 2,3,4 \}$ with $i \ne j$. Let $q \in \mathbb{C}$ satisfy $0 < |q| < 1$. Then each of the two fields $$ \mathbb{Q}( q, \theta_i(q), \theta_j(q), D\theta_k(q)) \quad\text{ and }\quad \mathbb{Q}( q, \theta_k(q), D\theta_k(q), D^2\theta_k(q)) $$ has transcendence degree $\ge 3$ over $\mathbb{Q}$

We know for any non-zero algebraic $q$ with $|q| < 1$, the three $\theta_k(q)$, in particular $\theta_2(q)$ is transcendental. Since

$$\sum_{n=1}^\infty 10^{-n(n+1)/2} = \frac{\sqrt[8]{10}}{2} \theta_2\left(\frac{1}{\sqrt{10}}\right) - 1$$

and using the fact $\frac{1}{\sqrt{10}}$ and $\frac{\sqrt[8]{10}}{2}$ are both algebraic, we find the number at hand is transcendental.