We investigate algebraic properties of the negative degree q-Bernstein bases. Our fundamental tool in this investigation is a recently introduced variant of the blossom, the multirational q-blossom, which provides the dual functionals for the negative degree q-Bernstein basis functions. By applying the dual functional property of the multirational q-blossom, we are readily able to generate several fundamental identities involving the negative degree q-Bernstein bases, including a new variant of Marsden’s identity, a partition of unity property, a reparametrization formula, and a formula for representing monomials. We also show how to use the homogeneous variant of the multirational q-blossom to convert between the q-Taylor bases and the negative degree q-Bernstein bases.
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Tuncer et al. (2025) studied this question.
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