The determinants of matrices with recursive entries (Q924319)

Let \((\alpha_{i}),(\beta_{i}),\dots,(\lambda_{i})\) be given sequences and define \(\Phi(i,j):=\delta_{i-1}\gamma_{j-1}-\nu_{i-1}\mu_{j-1}\); \(\Psi(i,j):=\delta _{i-1}\lambda_{j-1}+\nu_{i-1}\); and \(\Omega(i,j):= [\alpha_{i} -\Psi(i,1)\alpha_{i-1}] [\beta_{j}-\mu_{j-1}\beta_{j-1}]\). Now define the entries of the \(n\times n\) matrix \(A= [a_{ij}] \) recursively as follows: (1) \(a_{1i}:=\alpha_{1}\beta_{i}\) and \(a_{i1} :=\alpha_{i}\beta_{1}\) for \(i=1,\dots,n\); and (2) \(a_{ij}:=\mu_{j-1} a_{i,j-1}+\Phi(i,j)a_{i-1,j-1}+\Psi(i,j)a_{i-1,j}+\Omega(i,j)\) for \(i,j=2,\dots,n\). The main theorem of the paper is an expression for \(\det A\) in closed form. It generalizes some earlier results by the same authors. The proof uses an LU-factorization introduced by \textit{C. Krattenthaler} [Linear Algebra Appl. 411, 68--166 (2005; Zbl 1079.05008)].