2001/12/04 by Eric M. Rains, Rains, Eric M.
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Nonlinear Waves and Solitons #math.CO #math.QA
paper · pdf · doi:10.48550/arxiv.math/0112035
65 pages, LaTeX. v2-3: Minor corrections and additions (including teasers for the sequel). v4: C^+ notation changed to harmonize with the sequels (and more teasers added)
arxiv created 2004/02/07 · arxiv updated 2009/11/30
We consider two important families of BCn-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.