How to find the closed form of multiplications of exponentials?

Question

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.

Answer 1

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