I am looking for closed form of the sum:
$$\sum _{i=0}^{\infty } (-1)^i x^{\frac{1}{2} i (i k-k+2 n)}$$
we know the sum:
$$\sum _{i=0}^{\infty } (-1)^i x^{i k -k + n}=\frac{x^{n-k}}{x^k+1}$$
I was trying to reach the first sum using the second one, but no success. So any hint is appreciated.