# How to find the closed form of multiplications of exponentials?

I am trying to find a simplified formula for the following products:

$$(a^0+1)\cdot(a^1+1)\cdot(a^2+1)\cdot...\cdot(a^n+1)$$

but it seems quite difficult to figure out. Just wondering if anyone know how to do this.

• Generally speaking you won't get a closed form unless there is some nice cancellation. In this case consider that the coefficient of $a^k$ will be the number of ways you can write $k$ as a sum of distinct natural numbers up to and including $n$, multiplied by 2. Mar 23, 2018 at 22:44
• These coefficients are pretty well studied, so you should be able to find a resource on how to compute the number of partitions into distinct parts pretty easily. Mar 23, 2018 at 22:46
• Almost as a joke, at least for a simplified formula, this product is "just" $(-1;a)_{n+1}$ where appears the q-Pochhammer symbol. Mar 24, 2018 at 5:23

A closed form for the infinite product is the q-Pochhammer function : $$\prod_{n=0}^\infty(a^n+1)=(-1;a)_\infty$$ One can find some properties of this special function for example in : http://mathworld.wolfram.com/q-PochhammerSymbol.html
A more common closed form comes from the relationship with the Jacobi theta functions : $$\prod_{n=0}^\infty(a^n+1)=a^{-1/24}\frac{\vartheta_2(\frac{\pi}{6}\:,a^{1/3})}{\vartheta_2(\frac{\pi}{6}\:,a^{1/6})}$$ http://mathworld.wolfram.com/JacobiThetaFunctions.html