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:

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 \((T_n(q))_{n\ge 0}.\)

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：

除非\（0 \ le k \ le n \）和\（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} \）是\（\ 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} \）和\（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。 \）