Hermite Polynomials: What Are the Initial Values?

[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.

 

[fzero]

It might depend on whether you are reading a physics text or not: https://en.wikipedia.org/wiki/Hermite_polynomials#Definition If the text gives an explicit expression for any of them, you should be able to figure it out.

 

[cpsinkule]

It's Shankar's QM book. I'm having another difficulty as well. The recursion relation is
C_n+2=C_n(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.

 

[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 $N=5$.

 

[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.