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Defining a differential operator that acts on a non-commutative basis

Given a non-commutative basis ${x_0,x_1,x_2}$ I'd like to define a differential operator that acts as so $$ \Delta_i (\sum_{n=0}^\infty c_n x_i^n) = \sum_{n=1}^\infty c_n x_i^{n-1}, \quad \Delta_i x_ …
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