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This question is a follow-up of question A series that is rational? . Let $k=\mathbb F_q(T)$. Can one prove (or disprove) that the series $\sum_{n\ge0}(1-TX^{q^n})Y^{q^n}\in k[[X,Y]]$ is algebraic over $k(X,Y)$? In the mentioned link it is proved that the series does not belong to $k(X,Y)$.

If the series would be $\sum_{n\ge0}(1-X^{q^n})Y^{q^n}$, it is easy to prove...

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    $\begingroup$ The series is $f-Tg$ where $f=\sum_{n\geq0}Y^{q^n}$ and $g=\sum_{n\geq0}(XY)^{q^n}$. Both $f$ and $g$ are algebraic since $f^q-f=-Y$ and $g^q-g=-XY$. $\endgroup$
    – Laurent Moret-Bailly
    Commented Jul 11, 2020 at 16:52

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