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So, the canonical answer is the [q-Lucas theorem,][1]q-Lucas theorem, as pointed out in the comments.

This was proved in Olive, Gloria, Generalized powers, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.
[1]: https://www.math.upenn.edu/~peal/polynomials/q-analogues.htm#prelimQanaloguesQLucas

So, the canonical answer is the [q-Lucas theorem,][1] as pointed out in the comments.

This was proved in Olive, Gloria, Generalized powers, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.
[1]: https://www.math.upenn.edu/~peal/polynomials/q-analogues.htm#prelimQanaloguesQLucas

So, the canonical answer is the q-Lucas theorem, as pointed out in the comments.

This was proved in Olive, Gloria, Generalized powers, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.

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Per Alexandersson
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So, the canonical answer is the [q-Lucas theorem,][1] as pointed out in the comments.

This was proved in Olive, Gloria, Generalized powers, Am. Math. Mon. 72, 619-627 (1965). ZBL0215.07003.
[1]: https://www.math.upenn.edu/~peal/polynomials/q-analogues.htm#prelimQanaloguesQLucas