Relationships between identities for quantum Bernstein bases and formulas for hypergeometric series

Abstract

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-Saalschütz formula for hypergeometric series. The equivalence of the q-versions of these formulas and identities is also demonstrated.