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How can I prove the following equation: $\partial _q^{-1} f=\theta^{-1}(f)\partial_q^{-1}+\partial_q^{-1}\circ(\partial_q^*f)\circ \partial _q^{-1}$.

Where $\partial_q^{-1}$ is the formal inverse of the $q$-differential operator $\partial_q f(x)=\frac{f(qx)-f(x)}{(q-1)x}$, i.e. a integral operator, and $\theta$ is the $q$-shift operator,$\theta(f(x))=f(qx)$.

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Please provide more context, i.e. which book/page or which class/homework does this come from. –  David Mahone Jun 7 '13 at 5:44
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I don't know $q$-calculus, but is it legal to verify such an identity by post-composing with $\partial_q$, and applying the left and right side to an input like $\partial_q g$? –  S. Carnahan Jun 7 '13 at 7:05
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