Basic hypergeometric formulas and identities for negative degree q-Bernstein bases

Abstract

We utilize formulas for basic hypergeometric series to derive identities and formulas for negative degree q-Bernstein bases, including the Marsden identity, the partition of unity property, the monomial representation formula, the reparametrization formula, and the degree reduction formula. We show that all these identities are just special forms of the q-analogue of Gauss’ formula. We also provide a new proof for the q-analogue of Gauss’ formula by using the Marsden identity for negative degree q-Bernstein bases together with the identity theorem for analytic functions.