Asymmetric generalizations of the Filbert matrix and variants
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Four generalizations of the Filbert matrix are considered; with additional asymmetric parameter settings. Explicit formula 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.