Hall polynomials and basic hypergeometric series

Hall polynomials appear as the structure constants of Hall algebras
corresponding to Fq-linear categories. They can be thought of as a
generalization of the Gaussian polynomials (also known as q-binomial
coefficients), which count the number of k-dimensional subspaces of an
n-dimensional vector space over Fq. Algebraic identities in Hall algebras, such
as Green’s theorem, can be translated into statements about Hall polynomials,
which reduce to basic hypergeometric identities. I’ll go through a few examples
of these identities, as well as sketch a general theory for Dynkin quivers.