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Consider the standard Vandermonde $V(x_1, \ldots, x_n) = \prod_{i < j} (x_i - x_j)$. I am intersted in the calculation of the following expression for fixed $k$: $$\sum_i (x_i)^k (d/dx_i)^k V(x_1 , \ldots , x_n).$$ My guess is that it equals $c \cdot V(x_1, \ldots, x_n)$ where $c$ is an expression depending on $k$ and $n$ but not on the $x_i$'s. Is it true $c$? ? If yes, what is this constant $c$?

I think, if it is true, then it is pretty well-known. Would you be so kind to provide with the answer and/or proof and/or references? What can be the context people study it? Symmetric functions? Quantum Calogero-Moser?

# Vandermonde"V"Asuminvolving derivatives ,someidentity\sum_i(xi)^k(d/dxi)^kV=(????)*V-true?Whatis(???)ofVandermonde

Consider the standard Vandermonde V(x1, ... xn$V(x_1, \ldots, x_n) = prod \prod_{i < j} (xi x_i - xj), with i smaller jx_j)$. I am intersted in the calculation of the following expression for fixed k:

\sum_i $k$: $$\sum_i (xi)^k x_i)^k (d/dxi)^k V(x1 d/dx_i)^k V(x_1 , ... \ldots , xn) my x_n).$$ My guess is that it equals $c \cdot V(x_1, \ldots, x_n)$ where $c$ is equal to constant * V(x1 ... xn). Where "constant" some an expression depending on k,n, $k$ and $n$ but not on xithe $x_i$'s. Is it true ? $c$? If yes, what is this constant ?$c$?

I think(, if it is true) , then it is pretty well-known. Would you be so kind to provide with the answer and/or proof and/or references? What can be the context people study it? Symmetric functions? Quantum Calogero-Moser?

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# Vandermonde "V" derivatives, some identity \sum_i (xi)^k (d/dxi)^k V = (????) * V - true ? What is (???)

Consider the standard Vandermonde V(x1, ... xn) = prod (xi - xj), with i smaller j

I am intersted in the calculation of the following expression for fixed k:

\sum_i (xi)^k (d/dxi)^k V(x1 , ... , xn)

my guess that is equal to constant * V(x1 ... xn). Where "constant" some expression depending on k,n, but not on xi

Is it true ? If yes, what is this constant ?

I think (if it is true) it is pretty well-known. Would you be so kind to provide with the answer and/or proof and/or references ? What can be the context people study it ? Symmetric functions ? Quantum Calogero-Moser ?