in my endeavors I've stumbled upon the following identity: $$ \prod_{i=1}^N \left ( 1 + \frac{\partial}{\partial a_i}\right ) \left ( \Delta(a) \prod_{i=1}^N a_i \right ) = \Delta(a) \int_0^\infty dt e^{-t} \prod_{i=1}^N (t+a_i) $$ with Vandermonde determinant $\Delta (a) = \det \left ( a_j^{i-1} \right )_{i,j=1...N}$. I find it surprisingly hard to prove, maybe someone knows this formula or could suggest any way to derive it? I've tried to utilize Schur polynomials as the LHS is the $s_{(1_N)}$ but to no avail.
EDIT: My attempt of a proof started from the RHS where I use the generating functional of elementary symmetric functions:
$$ \prod_{i=1}^N (t + a_i ) = \sum_{n=0}^N \sigma_n (a) t^{N-n} $$ with elementary symmetric functions defined as $$ \sigma_n(a) = \sum_{1\leq i_1 < i_2 < ... < i_n \leq N} a_{i_1} a_{i_2} ... a_{i_n} $$ which gives $$ \text{RHS} = \Delta(a) \sum_{n=0}^N (N-n)!\sigma_n(a) $$