How the relationship $H_{2n+1}(x)=(-1)^n\sum_{s=0}^{n}(-1)^s(2x)^{2s+1}\cfrac{(2n+1)!}{(2s+1)!(n-s)!}$ is deduced ,working with Hermite polynomials ?
There is a problem so I don't know how to deduce the formula and it is that when I substitute $ 2n + 1 $ in the following relation:
\begin{align} H_n(x)&=\sum_{s=0}^{n/2}(-1)^s\cfrac{n!}{(n-2s)!s!}(2x)^{n-2s}\\ H_{2n+1}(x)&=\sum_{s=0}^{(2n+1)/2}(-1)^s\cfrac{(2n+1)!}{(2n+1-2s)!s!}(2x)^{2n+1-2s}\\ \end{align}
I get a sum whose upper limit is not a whole number.