An extension of the generalized pascal matrix and its algebraic properties

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