While reading about asymptotics of holonomic recurrences, I had the following impressions that are related to the divulgation of the theory:

  • I don't know what is the current status of the divulgation of the Poincaré-Perron theorem for second-order recurrences and its generalizations and extensions, and also its limitations. Sometimes a full asymptotic estimation is not possible, but this is intertwined with others arguments and things get confusing.
  • In the sense of generalizations, the work of Kooman for second-order recurrences is not too cited. The theorems have some restrictions, but sometimes this kind of recurrences appears and one can get a full asymptotic description.
  • The status of the Birkhoff-Trjitzinsky method is still unclear, and the mentions about its validity don't point to specific directions, because of the inherent complexity of the proofs (like in Wimp-Zeilberger paper). There has been some efforts to simplify the understanding of the theory like the Wong and Li papers on second-order recurrences, and the first paper of these has been widely cited, but I don't know to what extent this work is known in the community.

It would be possible to create an initiative to fully verify Birkhoff-Trjitzinsky? What are the main obstacles for this? Even more abstractly, do we need to go through this at some point? In another perspective, methods like the presented in Flajolet and Sedgewick book don't have an updated code for fast computation. Is this affecting also the divulgation of this method?

Very nice summaries of these techniques can be found in the web, see this thesis, this technical report or this book. Also papers like this one provide a good introduction that can lead to further search. I believe it would be nice if we can treat this theory here, so any thinking about the topic is welcome.

Note: This have been discussed for second-order recurrences here and also gives a sense of the importance of discussing it.

  • 4
    $\begingroup$ The paper arxiv.org/pdf/1502.06565.pdf by Garrabrant and Pak says "This result goes back to Birkhoff and Trjitzinsky (1932), and also Turrittin (1960). Although there are several gaps in these proofs, they are closed now, notably in [Imm]." That final citation is to "G. K. Immink, Reduction to canonical forms and the Stokes phenomenon in the theory of linear difference equations, SIAM J. Math. Anal. 22 (1991), 238–259." I personally know very little about this result, but I thought I would just point this out. $\endgroup$ Aug 8, 2021 at 2:44


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