Pascal-like determinants are recursive
Let \(P=[p_{i,j}]_{i,j\geq 0}\) be a Pascal-like triangle; i.e., an infinite matrix whose entries satisfy \(p_{i,j}=\lambda p_{i-1,j}+\mu p_{i,j-1}+\nu p_{i-1,j-1}\) for all \(i,j\geq 1\), and whose first column and row satisfy linear recurrences with constant coefficients of orders \(\rho\) and \(\sigma\), respectively. Let \(d_n\) be the \(n\)th principal minor, that is the determinant of the submatrix \([p_{i,j}]_{0\leq i,j\leq n}\). In this nice paper, the authors show that the sequence \((d_n)_{n\geq 0}\) satisfies a recurrence of the type \(d_{n}=\sum_{j=1}^{\delta} c_j \omega^{jn} d_{n-j}\), where \(c_j\) are constants, \(\omega=\lambda\mu +\nu\) and \(\delta={\rho+\sigma-2 \choose \rho-1}\). When \(\lambda=\mu=1\) and \(\nu=0\), then \(\omega=1\), therefore \((d_n)_{n\geq 0}\) satisfies a linear recurrence with constant coefficients of degree \(\delta\), which confirms a conjecture of \textit{R. Bacher} [J. Théor. Nombres Bordx. 14, 19--41 (2002; Zbl 1023.11011)]. The proof proceeds in two main steps. First it is shown that \(P\) can be transformed by minor-preserving transormations into a band-diagonal form, where a matrix \(P'=[p'_{i,j}]_{i,j\geq 0}\) is called \((r,s)\)-banded if \(p'_{i,j}= 0\) unless \(-s\leq i-j\leq r\), where \(r\) and \(s\) are fixed nonnegative integers. For the authors case, they show that one can take \(r=\rho-1\) and \(s=\sigma-1\) (Corollary 3). Secondly, they prove that if \(A\) is any \((r,s)\)-banded infinite matrix, then its principal minors satisfy a linear recurrence, with nonconstant coefficients, in general, of order \({r+s\choose r}\) (Theorem 2). The proof concludes by showing that for their case the coefficients of the above recurrence are geometrical progressions whose ratios are appropriate powers of \(\omega\) (Theorem 3). \noindent The paper is written in the language of generating functions and contains several numerical examples.
scientific article; zbMATH DE number 2131123