Generalizations of the Cauchy determinant (Q2754937)

Let \(A = (a_{ij})\) be the \(n\)-square matrix whose entries are \(a_{ij} = [P(y_j) + x_i Q(y_j)]/[1-x_iy_j]\) for polynomials \(P, Q\) and \(2n\) variables \(x_1,\ldots,x_n,y_1,\ldots,y_n.\) Furthermore, let \(\Omega(P,Q) = \det(A)\) and \(\Delta(x) = \prod_{1 \leq i < j \leq n} (x_i-x_j)\) (the Vandermonde determinant). Then the main result of this paper reads as follows: For two polynomials \(P\) and \(Q\) in one variable \(z\) with \(\text{deg}(zP+Q) > \text{deg}(P),\) let \(\varepsilon(P,Q) = 0,\) if \(\text{deg}(P) \geq \text{deg}(Q),\) and \(\varepsilon(P,Q) = L_c(Q)/L_c(zP+Q),\) if \(\text{deg}(P) < \text{deg}(Q),\) where \(L_c(P)\) denotes the leading coefficient of \(P.\) Suppose that \(zP+Q\) has \(m\) distinct roots \(\{ \beta_{\lambda} \}_{\lambda=1}^m\) such that \(z P(z) + Q(z) = L_c(zP+Q) \prod_{\lambda=1}^m (z-\beta_{\lambda}).\) Then we have the determinant evaluation formula NEWLINE\[NEWLINE\begin{multlined} \Omega(P,Q) = \frac{\Delta(x)\Delta(y)}{\prod_{1 \leq i,j \leq n} (1-x_iy_j)} \prod_{k=1}^n \{ y_k P(y_k) + Q(y_k) \} \times\\ \times \left\{ \varepsilon(P,Q) \prod_{i=1}^n x_i + L_c^{-1}(yP+Q) \sum_{l=1}^m \frac{P(\beta_l)}{\prod_{j \not= l} (\beta_l - \beta_j)} \prod_{i=1}^n \frac{1-x_i\beta_l}{y_i-\beta_l} \right\}.\end{multlined}NEWLINE\]NEWLINE By suitable choices of \(P, Q, \varepsilon(P,Q),\) and \(L_c(yP+Q),\) this result is applied to a variety of determinants. Particularly, if \(P \equiv 1\) and \(Q \equiv 0,\) then \(\Omega(P,Q)\) reduces to the Cauchy determinant NEWLINE\[NEWLINE\det \left[\frac{1}{1-x_iy_j} \right]_{1 \leq i,j \leq n} = \frac{\Delta(x)\Delta(y)}{\prod_{1 \leq i,j \leq n} (1- x_iy_j)} .NEWLINE\]