ASYMMETRIC GENERALIZATIONS OF THE FILBERT MATRIX AND VARIANTS

摘要

Four generalizations of the Filbert matrix are considered, with additional asymmetric parameter settings. Explicit formulae are derived for the LU-decompositions, their inverses, and the inverse matrix. The approach is mainly to use the $q$-analysis and to leave the justification of the necessary identities to the $q$-version of Zeilberger's algorithm for some of them, and for the rest of the necessary identities, to guess the relevant quantities and proving them later by induction.