Q-Analogue of The Pascal Matrix

Abstract

In this paper, \(q\)-analogues of the Pascal matrix and the symmetric Pascal matrix are studied. It is shown that the \(q\)-Pascal matrix \(\mathcal{P}_n\) can be factorized by special matrices and the symmetric \(q\)-Pascal matrix \(\mathcal{Q}_n\) has the LDU-factorization and the Cholesky factorization. As byproducts, some \(q\)-binomial identities are produced by linear algebra. Furthermore, these matrices are generalized in one or two variables, where a short formula for all powers of \(q\)-Pascal functional matrix \(\mathcal{P}_n[x]\) is given. Finally, it is similar to Pascal functional matrix, we have the exponential form for \(q\)-Pascal functional matrix.