AI 中文总结
该研究利用Pfaffian与Toeplitz相关公式，证明Baxter多项式序列具q-对数凸性，其q-精细Baxter数阵列行与列均为q-对数凹，相关结果可推广至d-Hoggatt数的q-模拟。
AI 中文摘要
我们利用Pfaffian子式求和公式与Jacobi-Trudi Toeplitz矩阵推导斜Schur函数中的Pfaffian展开式，由此得到若干含斜Schur函数乘积的生成函数的显式Schur展开。特别地，利用稀疏斜对称矩阵，我们给出了Narayana多项式q-对数凸性研究中产生的Schur正恒等式的Pfaffian证明。作为主要应用，我们证明Baxter多项式构成q-对数凸序列；进一步证明由精细Baxter数定义的Baxter变换保持对数凸性。最后，通过将q-精细Baxter数实现为矩形Schur函数的主 specialization，我们证明q-精细Baxter数阵列在每行和每列上均为q-对数凹的，这两个q-对数凹性结果可自然推广到d-Hoggatt数的q-模拟。
英文摘要
We use the Pfaffian minor summation formula together with Jacobi--Trudi Toeplitz matrices to derive Pfaffian expansions in skew Schur functions. This yields explicit Schur expansions for several generating functions involving products of skew Schur functions. In particular, using sparse skew-symmetric matrices, we provide a Pfaffian proof of a Schur-positive identity arising in the study of the $q$-log-convexity of the Narayana polynomials. As the main application, we prove that the Baxter polynomials form a $q$-log-convex sequence. We further show that the Baxter transformation defined by the refined Baxter numbers preserves log-convexity. Finally, by realizing the $q$-refined Baxter numbers as principal specializations of rectangular Schur functions, we prove that the array of $q$-refined Baxter numbers is $q$-log-concave both along each row and along each column. Both $q$-log-concavity results extend naturally to the $q$-analogues of the $d$-Hoggatt numbers.
Comments44 pages, 1 figure
