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Now I have equation $F(x) = x \sum_{k\ge 0} g_k F(x) F(qx) \cdots F(q^{k-1} x)$, I need to get the coefficient of $x^n$ in $F(x)$, can I apply $q$-Lagrange Inversion formula to this?

Moreover, I have another sequence $\{h_k\}$, I need to get the coefficient of $x^n$ in $\sum_{k\ge 0} h_k F(x) F(qx) \cdots F(q^{k-1} x)$.

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  • $\begingroup$ @Nemo How can I make the equation fit (3.1)? Or you actually mean theorem 5 will work? $\endgroup$
    – alpha1022
    Commented Mar 15, 2023 at 13:11

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