ZEILBERGER'S KOH THEOREM AND THE STRICT UNIMODALITY OF q-BINOMIAL COEFFICIENTS

摘要

A recent nice result due to I. Pak and G. Panova is the strict unimodality of the q-binomial coefficients ((a+b)(b))(q). Since their proof used representation theory and Kronecker coefficients, the authors also asked for an argument that would employ Zeilberger's KOH theorem. In this note, we give such a proof. Then, as a further application of our method, we also provide a short proof of their conjecture that the difference between consecutive coefficients of ((a+b)(b))(q) can get arbitrarily large, when we assume that b is fixed and a is large enough.