论文标题
三角复发，广义欧拉数和相关数字三角形
Triangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles
论文作者
论文摘要
许多组合和其他数字三角形是Graham-Knuth-Patashnik（GKP）类型的复发解决方案。分析研究了这种三角形及其定义复发。它们是由通过两个参与产生的转换组进行的：左右反射和上部二项式变换，在行动方面。该组还作用于三角形的双变量指数生成函数（EGF）。通过特征方法，任何GKP三角形的EGF在高斯超几何函数方面都具有隐式表示。有几个参数情况，可以以封闭形式获得此EGF。一个是当三角元素是hsu和shiue的普遍数字时。另一个是当它们是新定义的类型的尤拉利亚人数时。这些数字通过上二项式转换与Hsu-shiue相关，并且可以将其视为多项式碱基之间连接的系数，以概括经典的worpitzky身份。得出了许多涉及这些广义欧拉数和相关广义的纳拉亚纳数字的身份，包括在组合有意义的情况下进行封闭式评估。
Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted on by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.
