2008/10/13 by William Y. C. Chen, Robert L. Tang, Chen, William Y. C. +6
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #math.CO
paper · pdf · doi:10.48550/arxiv.0810.2247
29 pages
arxiv created 2008/10/13 · openalex publication_date 2008/10/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence \∑k=0nn\choose k2qk\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\choose k2\0≤ k≤ n is log-convexity preserving.