[ Gauss Polynomials $]

Find the sum $S_{l}(x)=g_{0, l}(x)-g_{1, l-1}(x)+g_{2, l-2}(x)-\ldots+(-1)^{l} g_{l, 0}(x)$.

The definition of Gauss polynomials $g_{k, 1}(x)$ can be found in the reference manual.
From problem $\underline{61522}$ v), it follows that $S_{l}(x)=\left(1-x^{l-1}\right) g_{0, l-1}(x)-\left(1-x^{l-2}\right) g_{1, l-2}(x)+\ldots+(-1)^{l-1}\left(1-x^{0}\right) g_{l-1,0}(x)$.

Applying the obvious equality

$\left(1-x^{m}\right) g_{k, m}(x)=\left(1-x^{k+m}\right) g_{k, m-1}(x)$ to each term in the last sum, we arrive at the equality $S_{l}(x)=(1-$ $\left.x^{l-1}\right) S_{l-2}(x)$.

Since $S_{1}(x)=0$, it follows from this that $S_{l}(x)=0$ for each odd $l$.

$S_{0}(x)=1$, therefore for even $l=2 n$ we get $S_{l}(x)=(1-x)\left(1-x^{3}\right) \ldots\left(1-x^{l-1}\right)=\frac{h_{2 n}(x)}{h_{n}\left(x^{2}\right)}$.

## Answer

$S_{2 n+1}(x)=0, \quad S_{2 n}(x)=(1-x)\left(1-x^{3}\right) \ldots\left(1-x^{2 n-1}\right)=\frac{h_{2 n}(x)}{h_{n}\left(x^{2}\right)}$.