Motivated by the classical Eulerian triangle and triangular arrays from staircase tableaux and tree-like tableaux, we study a generalized Eulerian array ${[T_{n,k}]}_{n,k\ge 0}$, which satisfies the recurrence relation:$$\begin{gathered} T_{n,k}=\lambda (a_{1}k+a_2)T_{n-1,k} \\ +[(b_1-d{a}_1)n-(b_1-2d{a}_1)k+b_2 \\ -d(a_1-a_2)]T_{n-1,k-1} \\ +\frac{d(b_1-d{a}_1)}{\lambda }(n-k+1)T_{n-1,k-2}, \end{gathered}$$ where $T_{0,0}=1$ and $T_{n,k}=0$ unless $0\le k\le n$. We derive some properties of ${[T_{n,k}]}_{n,k\ge 0}$, including the explicit formulae of $T_{n,k}$ and the exponential generating function of the generalized Eulerian polynomial $T_n(q)$, and the ordinary generating function of $T_n(q)$ in terms of the Jacobi continued fraction expansion, and real rootedness and log-concavity of $T_n(q)$, stability of the iterated Turán-type polynomial $T_{n+1}(q)T_{n-1}(q)-T_n^2(q)$. Furthermore, we also prove the q-Stieltjes moment property and 3-q-log-convexity of $T_n(q)$ and that the triangular convolution preserves Stieltjes moment property of sequences. In addition, we also give a criterion for γ-positivity in terms of the Jacobi continued fraction expansion. In consequence, we get γ-positivity of a generalized Narayana polynomial, which implies that of Narayana polynomials of types A and B in a unified manner. We also derive γ-positivity for a symmetric sub-array of ${[T_{n,k}]}_{n,k\ge 0}$, which in particular gives a unified proof of γ-positivity of Eulerian polynomials of types A and B.

Our results not only can immediately apply to Eulerian triangles of two kinds and arrays from staircase tableaux and tree-like tableaux, but also to segmented permutations and flag excedance numbers in colored permutations groups in a unified approach. In particular, we also confirm a conjecture of Nunge about the unimodality from segmented permutations.

受经典的欧拉三角形和阶梯形和树形形的三角阵列的影响，我们研究了广义的欧拉阵列 ${[{Ť}_{ñ，ķ}]}_{ñ，ķ\ge 0}$，它满足递归关系：$$\begin{gathered} {Ť}_{ñ，ķ}=\lambda （{\mathrm{一种}}_{\mathrm{1个}}ķ+{\mathrm{一种}}_2）{Ť}_{ñ-\mathrm{1个}，ķ} \\ +[（b_{\mathrm{1个}}-d{\mathrm{一种}}_{\mathrm{1个}}）ñ-（b_{\mathrm{1个}}-2d{\mathrm{一种}}_{\mathrm{1个}}）ķ+b_2 \\ -d（{\mathrm{一种}}_{\mathrm{1个}}-{\mathrm{一种}}_2）]{Ť}_{ñ-\mathrm{1个}，ķ-\mathrm{1个}} \\ +\frac{d（b_{\mathrm{1个}}-d{\mathrm{一种}}_{\mathrm{1个}}）}{\lambda }（ñ-ķ+\mathrm{1个}）{Ť}_{ñ-\mathrm{1个}，ķ-2}， \end{gathered}$$ 哪里 ${Ť}_{0，0}=\mathrm{1个}$ 和 ${Ť}_{ñ，ķ}=0$ 除非 $0\le ķ\le ñ$。我们推导了${[{Ť}_{ñ，ķ}]}_{ñ，ķ\ge 0}$，包括的明确公式 ${Ť}_{ñ，ķ}$ 和广义欧拉多项式的指数生成函数 ${Ť}_{ñ}（q）$，以及 ${Ť}_{ñ}（q）$ 就Jacobi而言，分数继续膨胀，并且根的真实生根和对数凹度 ${Ť}_{ñ}（q）$的Turán型多项式的稳定性 ${Ť}_{ñ+\mathrm{1个}}（q）{Ť}_{ñ-\mathrm{1个}}（q）-{Ť}_{ñ}^2（q）$。此外，我们还证明了q -Stieltjes矩性质和3- q -log-凸性${Ť}_{ñ}（q）$并且三角卷积保留了序列的Stieltjes矩特性。另外，我们还根据雅可比连续分数展开式给出了γ阳性的判据。结果，我们得到了广义Narayana多项式的γ-正性，这意味着A和B型Narayana多项式以统一的方式表示。我们还导出了的对称子阵列的γ-正性${[{Ť}_{ñ，ķ}]}_{ñ，ķ\ge 0}$，特别是给出了类型A和B的欧拉多项式的γ-正性的统一证明。

我们的结果不仅可以立即应用于阶梯形场景和树形场景的两种类型和阵列的欧拉三角形，而且可以统一地应用于彩色排列组中的分段排列和标志超出数。特别是，我们还证实了关于分段排列的单峰性的Nunge猜想。