A new multidimensional matrix inverse with applications to multiple q-series

We compute the inverse of a speciic innnite r-dimensional matrix, extending a matrix inverse of Krattenthaler. Our inversion is diierent from the r-dimensional matrix inversion recently found by Schlosser but generalizes a multidimensional matrix inversion previously found by Chu. As applications of our matrix inversion we derive some multidimensional q-series identities. Among these are q-analogues of Carlitz' multidimensional Abel-type expansion formulas. Furthermore, we derive a q-analogue of MacMahon's Master Theorem. 1. Introduction Matrix inversions are very important tools in combinatorics and special functions theory. In particular, it is a widely spread and often used method to derive and prove identities for (basic) hypergeometric series with the help of so-called \inverse relations" (see Section 4), which are immediate consequences of matrix inversions. (An inverse relation is in fact equivalent to its corresponding matrix inversion.) In order to be able to apply this method, explicit matrix inversions must be at hand. At this point it seems appropriate to elaborate a little on the history of (explicit) matrix inversions and inverse relations, in particular, since H. W. Gould's name is inevitably tied with it. Over time, people came across an increasing number of such explicit matrix inversions. In the 1960s, in his book 53], Riordan provided lists of known matrix inversions and, in fact, dedicated two complete chapters of his book to inverse relations and their applications. (Riordan's inverse relations were classiied and given a uniied method of proof by Egorychev 16].) A prominent part of these inverse relations were due to Gould, who studied them in a series of papers 27], 28], 29], 30]. This study culminated in the important discovery, jointly with Hsu, of a very