Hermite Polynomials

1. Jun 30, 2015

cpsinkule

I'm currently reading a text which uses Hermite polynomials defined in the recursive manner. The form of the polynomials are such that C0 C1 are the 0th and 1st terms of a taylor series that generate the remaining coefficients. The author then says the standard value of C1 and C0 are used, but he fails to mention what the standard is?!? Can anyone tell me what the initial values are? I found somewhere that the convention is 2n but I want to make sure.

2. Jun 30, 2015

fzero

3. Jun 30, 2015

cpsinkule

It's Shankar's QM book. I'm having another difficulty as well. The recursion relation is
Cn+2=Cn(2n+1-2ε)/(n+1)(n+2) where we added the constraint ε=(2n+1)/2. How can I generate any coefficients with this formula? For any N, the numerator is 0 because we had to choose ε that way in order for the series to terminate.

4. Jun 30, 2015

fzero

It's a confusing notation. It would be better to say that there must be some $n=N$ such that $\epsilon = (2N+1)/2$ and $C_{N+2}$ = 0. Then the $C_n$ with $n\leq N$ can be nonzero and determined by the recursion relation in terms of $C_0$ or $C_1$. You should try this out for some particular value like $N=5$.

5. Jun 30, 2015

cpsinkule

Thanks! I realized my mistake. I was thinking the ε was determined by the n in each term of the sum. I see now that the ε you choose for the nth hermite polynomial is the same for every term in the sum so that numerator in the recursion relationship is only 0 for the nth coefficient and up! It is a very confusing notation. He should have a different index for the specific ε than the dummy index in the sum. That's why I got confused.