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On a Stirling–Whitney–Riordan triangle
Journal of Algebraic Combinatorics ( IF 0.6 ) Pub Date : 2021-03-27 , DOI: 10.1007/s10801-021-01035-9
Bao-Xuan Zhu

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:

$$\begin{aligned} T_{n,k}= & {} (b_1k+b_2)T_{n-1,k-1}+[(2\lambda b_1+a_1)k+a_2+\lambda ( b_1+b_2)] T_{n-1,k}+\\&\lambda (a_1+\lambda b_1)(k+1)T_{n-1,k+1}, \end{aligned}$$

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 \(\mathbf{x} \)-totally positive with \(\mathbf{x} =(a_1,a_2,b_1,b_2,\lambda )\). 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 \(\mathbf{x} \)-Stieltjes moment property and 3-\(\mathbf{x} \)-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 \((T_{n,0})_{n\ge 0}\), we derive some properties similar to those of \((T_n(q))_{n\ge 0}.\)



中文翻译:

在斯特灵-惠特尼-里丹三角形上

基于第二类斯特林三角形,第二类惠特尼三角形和Riordan一个三角形,我们研究了Stirling–Whitney–Riordan三角形\([T_ {n,k}] _ {n,k} \)满足递归关系:

$$ \ begin {aligned} T_ {n,k} =&{}(b_1k + b_2)T_ {n-1,k-1} + [(2 \ lambda b_1 + a_1)k + a_2 + \ lambda(b_1 + b_2)] T_ {n-1,k} + \\&\ lambda(a_1 + \ lambda b_1)(k + 1)T_ {n-1,k + 1},\ end {aligned} $$

除非\(0 \ le k \ le n \)\(T_ {0,0} = 1 \),否则初始条件为\(T_ {n,k} = 0 \)。我们证明Stirling–Whitney–Riordan三角\([T_ {n,k}] _ {n,k} \)\(\ mathbf {x} \)-完全为正,\(\ mathbf {x} = (a_1,a_2,b_1,b_2,\ lambda)\)。我们证明行生成函数\(T_n(q)\)仅具有实零,而图兰型多项式\(T_ {n + 1}(q)T_ {n-1}(q)-T ^ 2_n (q)\)是稳定的。我们还给出了\(T_ {n,k} \)\(T_n(q)\)的指数生成函数的显式公式,并给出了Jacobi的普通生成函数的Jacobi连续分数展开式。\(T_n(q)\)。此外,我们得到\(\ mathbf {x} \)- Stieltjes矩属性和3- \(\ mathbf {x} \)- log-对凸凸度\(T_n(q)\)并显示了三角形卷积\ (z_n = \ sum _ {i = 0} ^ nT_ {n,i} x_iy_ {ni} \)保留序列的Stieltjes矩属性。最后,对于第一列\((T_ {n,0})_ {n \ ge 0} \),我们得出一些与\((T_n(q))_ {n \ ge 0}相似的属性。 \)

更新日期:2021-03-27
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