On a Stirling–Whitney–Riordan triangle

Abstract

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