On some Toeplitz matrices and their inversions

In this article, using the difference operator B(a[m]), we introduce a lower triangular Toeplitz matrix T   which includes several difference matrices such as Δ(1),Δ(m),B(r,s),B(r,s,t), and B(r̃,s̃,t̃,ũ) in different special cases. For any x ∈ w   and m∈N0={0,1,2,…}m∈N0={0,1,2,…}, the difference operator B(a[m  ]) is defined by (B(a[m])x)k=ak(0)xk+ak-1(1)xk-1+ak-2(2)xk-2+⋯+ak-m(m)xk-m,(k∈N0) where a[m] = {a(0), a(1), …, a(m)} and a(i) = (ak(i)) for 0 ⩽ i ⩽ m are convergent sequences of real numbers. We use the convention that any term with negative subscript is equal to zero. The main results of this article relate to the determination and applications of the inverse of the Toeplitz matrix T.