The q-Log-convexity of the Generating Functions of the Squares of Binomial Coefficients

TL;DR

This paper proves the q-log-convexity of generating functions for squared binomial coefficients, confirming a conjecture and establishing log-convexity preservation through combinatorial and algebraic methods.

Contribution

It confirms Liu and Wang's conjecture on q-log-convexity and shows that the associated linear transformation preserves log-convexity.

Findings

Proof of q-log-convexity for the polynomial sequence involving squared binomial coefficients

Verification that a key technical condition holds for these coefficients

Demonstration that the linear transformation preserves log-convexity

Abstract

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular array $\{{n\choose k}^2\}_{0\leq k\leq n}$ is log-convexity preserving.