Mathematics Asked by Kevin Lu on December 5, 2021
I’m wondering if there is a closed formula for the sum $a^1+a^4+a^9…$ and more generally $a^{1^n}+a^{2^n}+a^{3^n}…$ for real $a$ and $n$ such that $|a|<1$ and $n>1$.
According to Jacobi Triple product formula we have
$$prod_{m=1}^{infty}(1-x^{2m})(1+x^{2m-1}y^2)left(1+frac{x^{2m-1}}{y^2}right)=sum_{n=-infty}^{infty}x^{n^2}y^{2n}$$ for $|x|<1$ and $yneq0$. Then taking $x=a$ and $y=1$ we get $$prod_{m=1}^{infty}(1-a^{2m})(1+a^{2m-1}1^2)left(1+frac{x^{2m-1}}{1^2}right)=sum_{n=-infty}^{infty}a^{n^2}1^{2n}$$
$$impliesprod_{m=1}^{infty}(1-a^{2m})(1+a^{2m-1})^2=2sum_{n=1}^{infty}a^{n^2}+1$$
$$impliessum_{n=1}^{infty}a^{n^2}=frac{prod_{m=1}^{infty}(1-a^{2m})(1+a^{2m-1})^2-1}{2}$$
You can find more things here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.68.6437&rep=rep1&type=pdf
Answered by Shubhrajit Bhattacharya on December 5, 2021
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