Confluent \(q\)-extensions of some classical determinants (Q2575015)

Two celebrated determinants of Cauchy are extended and calculated. The first is a \(q,h\)-extension of the classical confluent extension of the Vandermonde determinant, where \(h\) is in the sense of the calculus of finite differences and the \(q\)-analogue of the number \(k\) is [\(k\)]:= \(k\) if \(q\) = 1 and \((1 - q^k)/(1 - q)\) elsewhere. The second is the double alternant.