An extension of slant Hankel operator, namely, the k-th-order λ-slant Hankel operator on the Lebesgue space \(L^{2}(\mathbb{T}^{n})\) , where \(\mathbb{T}\) is the unit circle and n ≥ 1, a natural number, is described in terms of the solution of a system of operator equations, which is subsequently expressed in terms of the product of a slant Hankel operator and a unitary operator. The study is further lifted in Calkin algebra in terms of essentially k-th-order λ-slant Hankel operators on \(L^{2}(\mathbb{T}^{n})\) .
