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 $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. 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. 
William Y. C. Chen (Nankai Univ.), Robert L. Tang (Nankai Univ.),   Larry X. W. Wang (Nankai Univ.) and Arthur L. B. Yang (N...