MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I wonder if there is anything known (formula, asymptotics, etc) of computing the remainder

$R_{k,m} \equiv H_{k} ~ \mod H_m$

for $k > m$, where $H_m$ denotes the $m$th Hermite polynomial (orthogonal under the weight $w(x) = e^{-x^2}$) and $\deg R_{k,m} \leq m-1$. I haven't been able to find anything online, neither could compute it through the recurrence relation of Hermite polynomials...


The motivation for my question is as follows. The $m$-point Gauss-quadrature is obtained by placing the nodes at the roots of $H_m$ and choosing the weights accordingly such that integrating any polynomial (with respect to weight $w$) of order $\leq 2m-1$ is exact. Now I want to know the error formula for polynomials of degree $k \geq m$, especially $H_k$. By computing $H_k$ modulo $H_m$, the integration error is given by the integration of the remainder $R_{k,m}$.

share|cite|improve this question
Why do you want to know this? And why ask this question about the Hermite polynomials? The thing is that Hermite polynomials have a lot of special properties, of which some they share with other polynomials / functions, but always only some. So it might be interesting which special properties you are interested in the proof using. – Helge Aug 16 '10 at 11:36
OK. I have added the motivation in the original post. – mr.gondolier Aug 16 '10 at 11:42 , particularly the formula in the upper right of that page. – J. M. Aug 16 '10 at 11:59
i am aware of this estimate. it does not help in this situation for $k > 2m$, because the derivative of $H_k$ is unbounded. Yes it is tight when $k=2m$ but in that case the integral can be easily calculated... – mr.gondolier Aug 16 '10 at 12:25
Wait... so what you want to do is to calculate the integral of $\exp\left(-x^2\right)H_K(x)$ with an m-point Gauss-Hermite formula? I don't understand why you wish to do this; it's not that hard to generate the nodes and weights of the Gauss-Hermite quadrature formula these days (e.g. via Golub-Welsch). – J. M. Aug 16 '10 at 13:50

Here is how to obtain a formula for $R_{k,m}$. Denote by $x_1, \dots, x_m$ the zeros of $H_m$. We have that $$ H_k(x) = P(x) H_{m}(x) + R_{k,m}(x) $$ for all $x$, so in particular for $x_j$ that $$ H_k(x_j) = R_{k,m}(x_j) $$ since $H_m(x_j) = 0$. So we know $R_{k,m}(x_j)$ at $m$ points, and it is a polynomial of degree $m-1$, so it is uniquely determined. In particular, one could write down a formula for it using some interpolation method.

I guess to understand this gadget, one would need to understand $H_k(x_j)$, which my first guess would be to check the already linked Abramowitz--Stegun book...

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.