A new extension of generalized Pascal-type matrix and their representations via Riordan matrix

The algebraic approach based on Pascal matrices is important in many fields of mathematics, ranging from algebraic geometry to optimization, matrix theory and combinatorics. The core of the proposed approach is to introduce a new family of Pascal-type matrices Ψ i , j , c , a [ x , y ] , x , y ∈ R - { 0 } with parameters c , a ∈ R + - { 1 } . By employing the effective matrix algebra tools, certain algebraic properties including the product formula, inverse matrix, determinant and eigen values are determined for the Pascal matrix Ψ i , j , c , a [ x , y ] . Further, some new families of matrices like the Fibonacci F i , j , c , a [ x , y ] , Lucas L i , j , c , a [ x , y ] , Pell S i , j , c , a [ x , y ] and other matrices are introduced and these are employed to derive factorization formulae for the Pascal matrix Ψ i , j , c , a [ x , y ] involving Riordan matrix. Finally, the properties and representations derived above for these matrices are further demonstrated for a matrix of particular order 3.