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.