论文标题
从指数的Riordan阵列中的总积极性
Total positivity from the exponential Riordan arrays
论文作者
论文摘要
Bender和Canfield证明了循环指数多项式的对数con症和几乎对数凸的性[J.组合。理论ser。 A 74（1996）]。 Schirmacher [J。组合。理论ser。 A 85（1999）]将它们扩展到$ q $ -log-concavity和几乎$ q $ -log-convexity。由这些动机，我们考虑了Toeplitz矩阵和Hankel矩阵的更强性能。 通过使用指数riordan阵列方法，我们给出了一些标准，以确定在多项式序列的普遍循环指标多项式的系数，Toeplitz矩阵和Hankel矩阵的总阳性，多项式序列的toeplitz矩阵和hankel矩阵，以指数式的式和较差的表单为差异。 最后，我们将我们的标准应用于满足一些复发关系的一些三角阵列，包括两种贝塞尔三角形及其概括，即lah三角形及其概括，同性的三角形和一些与二项式系数有关的三角形，Rook polynomials and Laguerre polynomialsials。我们不仅获得了这些低三角形的总积极性，以及$ q $ - stieltjes Moment Properties和$ 3 $ - $ Q $ -LOG-CONVEXITY的产生行为功能，而且还证明了他们的三角卷积保留了Stieltjes Moment属性。特别是，我们在$ q $ stieltjes Moment Property属于多项式的属性上解决了Sokal的猜想。
Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser. A 85 (1999)] extended them to $q$-log-concavity and almost $q$-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomial sequence in terms of the exponential formula, the logarithmic formula and the fractional formula, respectively. Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle and some triangles related to binomial coefficients, rook polynomials and Laguerre polynomials. We not only get total positivity of these lower-triangles, and $q$-Stieltjes moment properties and $3$-$q$-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve Stieltjes moment property. In particular, we solve a conjecture of Sokal on $q$-Stieltjes moment property of rook polynomials.
