We have been given a Hermite equation $ \frac{d^2 y}{dx^2} -2x \frac{dy}{dx}+2ny=0$
We need to use the Frobenius method to solve.
So far we have solved the indicial equation and got r = 0,1 and the recurrence relation as $$C_{m+2}=\frac{2(m+r-n)C_{m}}{(m+r+2)(m+r+1)}$$
We know that all odd m>0 are all vanishing terms.
The question i have is why the power series reduces to a polynomial, to a polynomial when r=1 and with n odd and n=1.