Some new applications of matrix inversions in \(A_r\) (Q1964789)

It was known from the literature that the matrices \((f_{nk})_{n,k=-\infty}^\infty\) and \((g_{kl})_{k,l=-\infty}^\infty\) are inverses of each other, where \[ f_{nk}={\prod _{j=k}^{n-1} (a_j+b_jc_k) \over \prod _{j=k+1}^{n} (c_j-c_k)},\qquad g_{kl}={(a_l+b_lc_l)\over (a_k+b_kc_k)} {\prod _{j=l+1}^{k} (a_j+b_jc_k)\over \prod _{j=l}^{k-1} (c_j-c_k)}. \] A multidimensional extension of this matrix inversion, associated to root systems, and their applications were found by the author of this paper in [\textit{M. Schlosser}, The Ramanujan J. 1, 243-274 (1997; Zbl 0934.33006)]. The purpose of this article is to give some new applications of the multidimensional matrix inversions. The author uses special cases of the multidimensional matrix inversions in conjunction with multiple basic hypergeometric summation theorems to derive a number of multidimensional \(q\)-series identities which themselves do not belong to the hierarchy of (basic) hypergeometric series. Among these are \(A_r\) terminating and nonterminating \(q\)-Abel and \(q\)-Rothe summation formulas. The author derives some identities of another type which appear to be new already in the one-dimensional case.