Generalized Pascal functional matrix and its applications (Q880029)

Let \(f(t;x)\) be a function of \(t\) with parameter \(x\) for which the \(n\)-th order derivatives with respect to \(t\) exist. The generalized Pascal functional matrix \({\mathcal P}_n[f(t;x)]\) is an \((n+1)\times(n+1)\) matrix such that \(({\mathcal P}_n[f(t;x)])_{ij}= {\binom ij}f^{(i-j)}(t;x)\) if \(i\geq j\) for \(i,j=0,1,2,\ldots,n\), and \(0\) otherwise. The authors show, if \({\mathcal P}_n[f(t;x)]\) and \({\mathcal P}_n[g(t;x)]\) are any two \((n+1)\times (n+1)\) Pascal functional matrices, then \({\mathcal P}_n[f(t;x)]{\mathcal P}_n[g(t;x)]={\mathcal P}_n[f(t;x)g(t;x)]={\mathcal P}_n[g(t;x)]{\mathcal P}_n[f(t;x)].\) They also prove some combinatorial identities, e.g. Tepper's identity, and they determine the \(LU\) decomposition of some well-known matrices.