Compute $\prod_{n=1}^{\infty} (1+x^{2n})$ for $|x|\lt 1$

Question

Compute for $|x|\lt 1$:$$\prod_{n=1}^{\infty} (1+x^{2n})$$

I'm having trouble computing this product. My work ends up contradicting and saying $x$ has to equal $1$. Anyone know how to do this?

Answer 1

Define:

$$Q_0:=\prod_{n>0}1-q^{2n},\;Q_1:=\prod_{n>0}1+q^{2n}, \;Q_2:=\prod_{n>0}1+q^{2n-1},\;Q_3:=\prod_{n>0}1-q^{2n-1}.$$

There are many standard identities between theta functions and infinite products such as $Q_0,Q_1,Q_2,Q_3.$ For example, $1=Q_1Q_2Q_3$ and DLMF equations 20.4.4 and 20.4.5.

The infinite product you want is $\;Q_1.\;$ To compute it via the q-Pochhammer symbol use $\;Q_1 = (q^2,-q^2)_\infty = 1/(q^2,q^4)_\infty.\;$ The OEIS sequence A000009 information is of interest.

Answer 2

Note that

\begin{align*} \prod_{1 \leq k} \left( 1 + z^k \right) &= \left( 1 + z^1 \right)\left( 1 + z^2 \right) \cdots \left( 1 + z^k \right) \cdots \\ &= \sum_{0 \leq n < \infty} f(n) z^n \quad \left| \quad f(n) = \left[ \begin{array}{l} \text{N. of partit. of $n$ into distinct parts} \\ \text{N. of partit. of $n$ into odd parts} \end{array} \right. \right. \end{align*}

and unfortunately, the N. of partitions, whether as per standard definition or into distinct / odd parts, do not have any known closed generating function.