Is there a function with this Taylor series?

Question

Is there a function with a power series like this? $$S(x)=\sum_{k=1}^\infty(-1)^k\left(x^{\frac{k(3k-1)}2}+x^{\frac{k(3k+1)}2}\right)$$

I tried differentiating and hoping it simplifies into a know sum with a closed form, but it doesn't.

Answer 1

There is a function Jacobi Theta:

$$ \sum_{k=1}^{\infty}(-1)^k x^{k^2}=\frac{1}{2}(\theta_4(x)-1), x<1 $$

Make a substitution $y=\sqrt{x}$

$$S(y)=\sum_{k=1}^{\infty}(-1)^k (y^{k(3k-1)}+y^{k(3k+1)}) = \sum_{k=1}^{\infty}(-1)^k y^{3k^2} (\frac{1}{y}+y)=\frac{1}{2}(\theta_4(y^3)-1)(\frac{1}{y}+y)$$

$$S(x)=\frac{1}{2}(\theta_4(x^{\frac{3}{2}})-1)(\frac{1+x}{\sqrt{x}}), x^3<1$$