@ARTICLE{Krattenthaler96anew, author = {C. Krattenthaler}, title = {A New Matrix Inverse}, journal = {Proc. Amer. Math. Soc}, year = {1996}, volume = {124}, pages = {47--59} }

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Abstract

. We compute the inverse of a specific infinite-dimensional matrix, thus unifying a number of previous matrix inversions. Our inversion theorem is applied to derive a number of summation formulas of hypergeometric type. 1. Introduction. Let F = (f nk ) n;k2Z (Zdenotes the set of integers) be an infinitedimensional lower-triangular matrix; i.e. f nk = 0 unless n k. The matrix (f \Gamma1 kl ) k;l2Z is the inverse matrix of F if and only if X nkl f nk f \Gamma1 kl = ffi nl for all n; l 2 Z. Such matrix inversions are very important in many fields of combinatorics and special functions. For example, when dealing with combinatorial sums, application of the so-called "inverse relations" (see (4.1) and (4.2)), which base on matrix inversion, helps to simplify problems, or yields new identities. Riordan dedicated two chapters of his book [21] to inverse relations and its applications. Riordans inverse relations were classified and given a unified method of proof by Egorychev [7, ch.3]...