Some results of the generalized Jacobsthal sequence and the generalized Jacobsthal–Lucas sequence

In this paper, we investigate two classes of sequences: the generalized Jacobsthal sequence (JFn,p)n≥0 and the generalized Jacobsthal–Lucas sequence (JLn,p)n≥1, which are defined, respectively by JFn,p=∑k=0⌊n/(1+p)⌋JFp(n,k),n=0,1,2,…and JLn,p=∑k=0⌊n/(1+p)⌋JLp(n,k),n=1,2,3,…,where JFp(n,k)=∑i=k⌊n/(1+p)⌋(n−ipi)(ik),JLp(n,k)=∑i=k⌊n/(1+p)⌋nn−ip(n−ipi)(ik).We call polynomials JFn,p(x)=∑kJFp(n,k)xkandJLn,p(x)=∑kJLp(n,k)xkthe generalized Jacobsthal polynomials and the generalized Jacobsthal–Lucas polynomials, respectively. For p = 1 and x = 1, we obtain the Jacobsthal sequence (JFn,1(1))n≥0 and the Jacobsthal–Lucas sequence (JLn,1(1))n≥1. We shall prove the list of properties of the generalized Jacobsthal sequences and the generalized Jacobsthal–Lucas sequences. We are going to exhibit the real-rootedness of the generalized Jacobsthal polynomials and the generalized Jacobsthal–Lucas polynomials, the total positivity of coefficient matrices and the asymptotic normality of the coefficients of the generalized Jacobsthal polynomials and the generalized Jacobsthal–Lucas polynomials.