On an explicit formula for inverse of triangular matrices
Pinakadhar Baliarsingh, Salila Dutta · Journal of the Egyptian Mathematical Society · 2014
In the present article, we define difference operators B L ( a [ m ] ) and B U ( a [ m ] ) which represent a lower triangular and upper triangular infinite matrices, respectively. In fact, the operators B L ( a [ m ] ) and B U ( a [ m ] ) are defined by ( B L ( a [ m ] ) x ) k = ∑ i = 0 m a k - i ( i ) x k - i and ( B U ( a [ m ] ) x ) k = ∑ i = 0 m a k + i ( i ) x k + i for all k , m ∈ N 0 = { 0 , 1 , 2 , 3 , … } , where a [ m ] = { a ( 0 ) , a ( 1 ) , … a ( m ) } , the set of convergent sequences a ( i ) = ( a k ( i ) ) k ∈ N 0 ( 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 ) , Δ m , Δ ( m ) ( m ∈ N 0 ) , Δ α , Δ ( α ) ( α ∈ R ) , B ( r , s ) , B ( r , s , t ) , B ( r ̃ , s ̃ , t ̃ , u ̃ ) , and many others. Also, we derive an alternative method for finding the inverse of infinite matrices B L ( a [ m ] ) and B U ( a [ m ] ) and as an application of it we implement this idea to obtain the inverse of triangular matrices with finite support.