A short proof of Krattenthaler formulas
type: Item
instance of: scholarly article
title: A short proof of Krattenthaler formulas
zbMATH Open document ID: 1020.15004
author: Xinrong Ma
publication date: 2003-09-25T00:00:00Z
review text: 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.
reviewed by: David Garber
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 05A19
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 1885828
zbMATH Keywords: matrix inversion
zbMATH Keywords: \(q\)-analogue
zbMATH Keywords: Lagrange interpolation
zbMATH Keywords: Krattenthaler formula
zbMATH Keywords: infinite-dimensional lower-triangular matrices
MaRDI profile type: Publication
full work available at URL: s101140200161
OpenAlex ID: W2009434690
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DOI: 10.1007/S101140200161
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 1885828