An extension of the generalized pascal matrix and its algebraic properties

Abstract

The extended generalized Pascal matrix can be represented in two different ways: as a lower triangular matrix Φ n [ x, y ] or as a symmetric Ψ n [ x, y ]. These matrices generalize P n [ x ], Q n [ x ], and R n [ x ], which are defined by Zhang and by Call and Velleman. A product formula for Φ n [ x, y ] has been found which generalizes the result of Call and Velleman. It is shown that not only can Φ n [ x, y ] be factorized by special summation, but also Ψ n [ x, y ] as Q n [ xy ]Φ s T [ y ,1/ x ] or Φ n [ x, y ] P n T [ y/x ]. Finally, the inverse of Ψ n [ x, y ] and the values of det Φ n [ x, y ], det Φ n −1 [ x, y ], det Ψ n [ x, y ], and det Ψ n −1 [ x, y ] are given.