Numerical Triangles and Several Classical Sequences

In 1991 Ferri, Faccio and D’Amico introduced and investigated two numerical triangles, called the DFF and DFFz triangles. Later Trzaska also considered the DFF triangle. And in 1994 Jeannin generalized the two triangles. In this paper, we focus our attention on the generalized Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas polynomials, and several numerical triangles deduced by them. 1. THE GENERALIZED FIBONACCI AND LUCAS POLYNOMIALS Let us define a sequence of polynomials {Fn(y)} by the recurrence relation Fn+1(y) = Fn(y) + yFn−1(y), n ≥ 1, (1.1) where F0(y) = a, F1(y) = b. Notice that (1.1) yield the Fibonacci and Lucas sequences Fn and Ln when y = 1 with the initial values a = b = 1, a = 2, b = 1, respectively. Define Fn(y) = n ∑ k=0 fn,ky , F (x, y) = ∑ n≥0 Fn(y)x. (1.2) By (1.1) and (1.2) it is easy to derive F (x, y) = a+ (b− a)x 1− x− x2y , (1.3) and fn,k = [xy]F (x, y) = [xy](a+ (b− a)x) ∑ k≥0 xy (1− x)k+1 = a ( n− k n− 2k ) + (b− a) ( n− k − 1 n− 2k − 1 ) = a ( n− k − 1 k − 1 ) + b ( n− k − 1 k ) , which satisfies the recurrence fn+1,k = fn,k + fn−1,k−1, with the initial conditions f0,0 = a, f1,0 = b. Let an+1,k = f2n+2,n−k+1 and bn,k = f2n+1,n−k, then we have an+1,k = a ( n+ k 2k ) + b ( n+ k 2k − 1 ) , bn,k = a ( n+ k 2k + 1 ) + b ( n+ k 2k ) ,