On a Stirling–Whitney–Riordan triangle.

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: T n , k = (b 1 k + b 2) T n - 1 , k - 1 + [ (2 λ b 1 + a 1) k + a 2 + λ (b 1 + b 2) ] T n - 1 , k + λ (a 1 + λ b 1) (k + 1) T n - 1 , k + 1 , where initial conditions T n , k = 0 unless 0 ≤ k ≤ n and T 0 , 0 = 1 . 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 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 n 2 (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 x -Stieltjes moment property and 3- x -log-convexity of T n (q) and show that the triangular convolution z n = ∑ i = 0 n T n , i x i y n - i preserves Stieltjes moment property of sequences. Finally, for the first column (T n , 0) n ≥ 0 , we derive some properties similar to those of (T n (q)) n ≥ 0. [ABSTRACT FROM AUTHOR]