On a generalized Jordan form of an infinite upper triangular matrix

Jordan Canonical Forms of Riordan Arrays

“…From the first equation of (5) and assumption that the transition matrix X is a proper Riordan array (i.e. x 0 = 0 and y 1 = 0) one concludes that d 0 = α.…”

“…In [6] the authors show how to compute some functions of truncated Riordan arrays using their Jordan canonical form. For the sake of clarity, let us present the definition: the triangular matrix of the form J R = X −1 RX, with X being a Riordan array, that is a direct sum of Jordan cells (possibly also infinite) is called the Jordan form of Riordan array R. If we allowed X be any N 0 × N 0 matrix with a finite number of nonzero entries in each row, then the generalized Jordan form would for sure exist [5]. However, would they…”