Combinatorics: Product Rules.

Question

I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:

I have the homogeneous polynomial function $$f(X) = \sum_{u_1, \dots u_n = 1}^n X_{u_1} \cdots X_{u_n}$$ where $n$ is even, and the differential operator $$ L = \sum_{j=1}^n \lambda_j \frac{\partial^2}{\partial X_j^2},$$ where $\lambda_j$ are some nonzero numbers.

Problem: Calculate $L^{n/2} f(X)$. Obviously, this is constant and of the form $$ L^{n/2} f(X) = C(n) \sum_{u_1, \dots u_{n/2}=1}^{n/2} \lambda_{u_1} \cdots \lambda_{u_{n/2}},$$ for some number $C(n)$. But what is $C(n)$?

Answer 1

It seems to me that in either your expression for $L$ or for $L^{n/2}f(X)$, you need to replace $\lambda_j$ with $1/\lambda_j$. (I see this has been corrected, so my previous sentence is irrelevant.) In any event, since $f(X)=(X_1+\cdots+X_n)^n$ it is clear that $C(n)=n!$.