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

We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence { ∑n k=0 ( n k )2 q}n≥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 k )2 }0≤k≤n is log-convexity preserving.