Left quotient operator on regular expressions

Given two regular languages $L_1$ and $L_2$, we can define the left language quotient operations as $L_2\backslash_lL_1 := \{ v \ \vert \ \exists u, uv \in L_1 \land u \in L_2 \}$. The Brzozowski derivative is a special case of the left quotient by a singleton language: $\{c\}\backslash_lL_1 := \{ v \ \vert \ cv \in L_1 \}$. This leads to the following question: is there a purely inductive definition of the left quotient operation on regular expressions? i.e. a function $f : RE \times RE \to RE$ s.t. $\mathcal{L}(f(R_2,R_1))=\mathcal{L}(R_2)\backslash_l\mathcal{L}(R_1)$.