On a Stirling-Whitney-Riordan triangle

Abstract

Motivated by the Stirling triangle of the second kind, the Whitney triangle of the second kind and a 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 ≤ k ≤ n and T 0,0 = 1. Let its rowgenerating function T n (q) = k≥0 T n,k q k for n ≥ 0.We prove that the Stirling-Whitney-Riordan triangle [T n,k ] n,k is x-totally positive with x = (a 1 , a 2 , b 1 , b 2 , λ). We show real rootedness and log-concavity of T n (q) and stability of the Turán-type polynomial T n+1 (q)T n−1 (q)−T 2 n (q). We also present explicit formulae of T n,k and the exponential generating function of T n (q), and the ordinary generating function of T n (q) in terms of a Jacobi continued fraction expansion. Furthermore, we get the x-Stieltjes moment property and 3-x-log-convexity of T n (q) and that the triangular convolution z n = n i=0 T n,k x i y n−i preserves Stieltjes moment property of sequences. Finally, for the first column (T n,0 ) n≥0 , we derive some similar properties to those of (T n (q)) n≥0 .