The polynomial ring $B_r:=mathbb{Q}[e_1,ldots,e_r]$ in $r$ indeterminates is a representation of the Lie algebra of all the endomorphism of $mathbb{Q}[X]$ vanishing at powers $X^j$ for all but finitely many $j$. We determine a $B_r$-valued formal power series in $r+2$ indeterminates which encode the images of all the basis elements of $B_r$ under the action of the generating function of elementary endomorphisms of $mathbb{Q}[X]$, which we call the structural series of the representation. The obtained expression implies (and improves) a formula by Gatto & Salehyan, which only computes, for one chosen basis element, the generating function of its images. For sake of completeness we construct in the last section the $B=B_infty$-valued structural formal power series which consists in the evaluation of the vertex operator describing the bosonic representation of $gl_{infty}(mathbb{Q})$ against the generating function of the standard Schur basis of $B$. This provide an alternative description of the bosonic representation of $gl_{infty}$ due to Date, Jimbo, Kashiwara and Miwa which does not involve explicitly exponential of differential operators.
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