Log-concavity results for a biparametric and an elliptic extension of the $q$-binomial coefficients

Abstract

We establish discrete and continuous log-concavity results for a biparametric extension of the $q$-numbers and of the $q$-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Turán's inequality.

17 pages; dedicated to Bruce Berndt, on the occasion of his 80th birthday; minor changes; to appear in the International Journal of Number Theory