A characterization of inverse relations

AbstractWe characterize all pairs F = (F(n,k)) and G = (G(n,k)) of inverse infinite, lower-triangular matrices by a ‘dual’ pair of recurrence relations that their entries F(n,k) and G(n,k) must satisfy. This characterization provides a unified approach towards all inverse relations based on such inversion problems. The computation of the inverse of a given infinite, lower-triangular matrix F is reduced to finding a recurrence of the required form that its entries must satisfy. The inverse matrix G is then determined in a similar way by the dual recurrence. We provide historical motivation, and then use our characterization theorem to invert a number of important infinite, lower-triangular matrices. These include matrix inversions of Gould and Hsu, Krattenthaler, Carlitz, Bressoud, as well as Andrews' matrix formulation of the Bailey Transform. These examples illustrate how shift operators or summation theorems such as the q-Chu-Vandermonde summation are used to help find recurrence relations of the required type for the entries of F and G