Abstract
In this note, we shall give conditions which guarantee that \(\frac{1-q^b}{1-q^a}\Big [\begin{array}{l} n\\ m\\ \end{array}\Big ]\in \mathbb {Z}[q]\) holds. We shall provide a full characterisation for \(\frac{1-q^b}{1-q^a}\Big [\begin{array}{l} ka\\ m\\ \end{array}\Big ]\in \mathbb {Z}[q]\). This unifies a variety of results already known in literature. We shall prove new divisibility properties for the binomial coefficients and a new divisibility result for a certain finite sum involving the roots of the unity.
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This research was supported by UAEU/COS Grant IRG-17/15.
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Bachraoui, M.E. Towards characterising polynomiality of \(\frac{1-q^b}{1-q^a}{n\brack m}\) and applications. Ramanujan J 45, 291–298 (2018). https://doi.org/10.1007/s11139-016-9877-y
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DOI: https://doi.org/10.1007/s11139-016-9877-y