A short proof of Krattenthaler formulas
The author gives a short proof of a special case of a result due to \textit{C. Krattenthaler} [Proc. Am. Math. Soc. 124, 47-59 (1996; Zbl 0843.15005)], which can be stated as follows: Let \(F\) and \(G\) be two infinite-dimensional lower-triangular matrices defined by \[ f_{n,k} = \frac{\prod_{i=k}^{n-1}(a_i+b_k)}{\prod_{i=k+1}^n (b_i-b_k)}\qquad\text{and}\qquad g_{n,k} = \frac{a_k+b_k}{a_n+b_n}\frac{\prod_{i=k+1}^n (a_i+b_n)}{\prod_{i=k}^{ n-1} (b_i-b_n)} \] respectively (where \(a_i\) and \(b_i\) are complex numbers such that \(b_i \not = b_j\) for \(i \not = j\)), then the matrices \(G\) and \(H\) are inverse to each other. The author uses the Lagrange interpolation for approximating polynomials in his proof. He also shows that this form of invertible matrices, given by such equations for the entries of \(F\) and \(G\), is uniquely determined.
infinite-dimensional lower-triangular matrices
scientific article; zbMATH DE number 1885828
scientific article