On an explicit formula for inverse of triangular matrices

In the present article, we define difference operators BL(a[m])BL(a[m]) and BU(a[m])BU(a[m]) which represent a lower triangular and upper triangular infinite matrices, respectively. In fact, the operators BL(a[m])BL(a[m]) and BU(a[m])BU(a[m]) are defined by (BL(a[m])x)k=∑i=0mak-i(i)xk-i and (BU(a[m])x)k=∑i=0mak+i(i)xk+i for all k,m∈N0={0,1,2,3,…}k,m∈N0={0,1,2,3,…}, where a[m]={a(0),a(1),…a(m)}a[m]={a(0),a(1),…a(m)}, the set of convergent sequences a(i)=(ak(i))k∈N0(0⩽i⩽m)a(i)=(ak(i))k∈N0(0⩽i⩽m) of real numbers. Indeed, under different limiting conditions, both the operators unify most of the difference operators defined by various triangles such as Δ,Δ(1),Δm,Δ(m)(m∈N0),Δα,Δ(α)(α∈R),B(r,s),B(r,s,t),B(r̃,s̃,t̃,ũ), and many others. Also, we derive an alternative method for finding the inverse of infinite matrices BL(a[m])BL(a[m]) and BU(a[m])BU(a[m]) and as an application of it we implement this idea to obtain the inverse of triangular matrices with finite support.