Relationships Between Identities for Quantum Bernstein Bases and Formulas for Hypergeometric Series

Two seemingly disparate mathematical entities - quantum Bernstein bases and hypergeometric series - are revealed to be intimately related. The partition of unity property for quantum Bernstein bases is shown to be equivalent to the Chu-Vandermonde formula for hypergeometric series, and the Marsden identity for quantum Bernstein bases is shown to be equivalent to the Pfaff-Saalschutz formula for hypergeometric series. The equivalence of the q-versions of these formulas and identities is also demonstrated.