论文标题
在斯特林 - 惠特尼 - 里亚人三角上
On a Stirling-Whitney-Riordan triangle
论文作者
论文摘要
基于第二类的斯特林三角，第二类的惠特尼三角和一个三角形的三角形，我们研究了一个stirling-whitney-riordan三角形$ [t_ {n，k} _ { t_ {n，k}＆=＆（b_1k+b_2）t_ {n-1，k-1}+[（2λB_1+a_1）k+a_2+λ（b_1+b_2）] \ end {eqnarray*}，其中初始条件$ t_ {n，k} = 0 $，除非$ 0 \ le k \ le n $和$ t_ {0,0,0} = 1 $。 我们证明了Stirling-Whitney-Riordan Triangle $ [t_ {n，k}] _ {n，k} $是$ \ textbf {x} $ - 完全呈$ \ textbf {x x} =（a_1，a_1，a_2，a_2，a_2，b_1，b_1，b_2，λ）$。我们表明，生成的函数$ t_n（q）$只有真实的零，而turán-type多项式$ t_ {n+1}（q）t_ {n-1}（q）-t^2_n（q）$稳定。我们还提供了$ t_ {n，k} $的显式公式以及$ t_n（q）$的指数生成函数，并为$ t_n（q）$的普通生成函数提供jacobi的持续分数扩展。此外，我们得到$ \ textbf {x} $ - stieltjes moment属性和$ 3 $ - $ \ textbf {x} $ - $ t_n（q）$的log-convexity $ t_n（q）$，并显示三角卷积$ z_n = \ sum__序列的stieltjes瞬间属性。最后，对于第一列$（t_ {n，0}）_ {n \ geq0} $，我们得出了一些类似于$（t_n（q））_ {n \ geq0}的属性。
Based on the Stirling triangle of the second kind, the Whitney triangle of the second kind and one triangle of Riordan, we study a Stirling-Whitney-Riordan triangle $[T_{n,k}]_{n,k}$ satisfying the recurrence relation: \begin{eqnarray*} T_{n,k}&=&(b_1k+b_2)T_{n-1,k-1}+[(2λb_1+a_1)k+a_2+λ( b_1+b_2)] T_{n-1,k}+\\ &&λ(a_1+λb_1)(k+1)T_{n-1,k+1}, \end{eqnarray*} where initial conditions $T_{n,k}=0$ unless $0\le k\le n$ and $T_{0,0}=1$. We prove that the Stirling-Whitney-Riordan triangle $[T_{n,k}]_{n,k}$ is $\textbf{x}$-totally positive with $\textbf{x}=(a_1,a_2,b_1,b_2,λ)$. We show that the row-generating function $T_n(q)$ has only real zeros and the Turán-type polynomial $T_{n+1}(q)T_{n-1}(q)-T^2_n(q)$ is stable. We also present explicit formulae for $T_{n,k}$ and the exponential generating function of $T_n(q)$ and give a Jacobi continued fraction expansion for the ordinary generating function of $T_n(q)$. Furthermore, we get the $\textbf{x}$-Stieltjes moment property and $3$-$\textbf{x}$-log-convexity of $T_n(q)$ and show that the triangular convolution $z_n=\sum_{i=0}^nT_{n,i}x_iy_{n-i}$ preserves Stieltjes moment property of sequences. Finally, for the first column $(T_{n,0})_{n\geq0}$, we derive some properties similar to those of $(T_n(q))_{n\geq0}.$
