The Markov-WZ Method

Andrei Markov’s 1890 method for convergence-acceleration of series bears an amazing resemblance to WZ theory, as was recently pointed out by M. Kondratieva and S. Sadov. But Markov did not have Gosper and Zeilberger’s algorithms, and even if he did, he wouldn’t have had a computer to run them on. Nevertheless, his beautiful ad-hoc method, when coupled with WZ theory and Gosper’s algorithm, leads to a new class of identities and very fast convergence-acceleration formulas that can be applied to any infinite series of hypergeometric type. Preface: Where are the Examples? One Example is given in the Appendix, but lots and lots of examples can be found at http://www.math.rutgers.edu/ ̃zeilberg/Markov/Examples.html. Many other interesting examples for acceleration formulas will be found in a forthcoming article by the first author[Mo1]. In addition, readers can generate (potentially) infinitely many other examples by using the Maple package MarkovWZ downloadable from that page. No knowledge of Maple is necessary!, just a basic knowledge of typing! Read the on-line instructions! The research (and of course, writing the Maple packages) is joint work between the authors. The actual writing was done by the second author. In particular we means the Royal we. How Andrei Andreyevich Markov (almost) Discovered WZ Pairs On Jan. 7, 2004, one of us (DZ) received an E-mail message from Sergei Sadov, that he kindly allowed us to quote from. * Thanks to Herb Wilf, the referee, Ben Kalba, and Tipesh Gadol for many suggestions the electronic journal of combinatorics 11 (2004), #R53 1 > From [email protected] Wed Jan 7 21:38:58 2004 To: Doron Zeilberger cc: Margo Kondratieva Subject: Markov’s transformation of series Dear Professor Zeilberger, I am finally writing to you, which is what I ought to have done at least 16 months ago, after narrowly missing you at ICM in Beijing. At the ICM-2002 we, I and my wife and coauthor Margo (I speak for both of us), presented a poster under the title "Summation technique forgotten for a century: Markov (1890) Wilf-Zeilberger (1990)". The abstract can be found on p. 404 of the ICM Abstracts volume. [In my present view, some particulars in that little text are inaccurate.] Moving to a new place, teaching, not making progress in the topic -resulted in a delay, which grew indefinitely. This summer, visiting Markov’s home city St.Petersburg (Russia), and combing through old textbooks on series during this new year break, we have collected more material around Markov’s memoir of 1890, aiming to answer the question why it had been forgotten. What Markov did in his memoir, very much resembles the WZ approach. Currently we are preparing a first larger-than-abstract size publication. We’d like to hear your firsthand opinion on Markov’s work. You can find the scanned pdf’s of Markov’s memoir at http://www.math.mun.ca/~sergey/Research/History/Markov/markov1890.html . A particular question, which, perhaps, you can answer easily, is: whether Sister Celine had cited Markov? I enjoyed reading some (since I didn’t read all) of your Opinions and your opinion on this subject will be greatly appreciated here!