By George Spahn and Doron Zeilberger
Written: June 21, 2023 ; Last update of this web-page: Nov. 24, 2023
Appeared in J. Difference Equations and Applications [Special Elaydi issue] (on-line before print, volume and page tbd)
We demonstrate the power of Experimental Mathematics and Symbolic Computation to study intriguing problems on rational difference equations, studied extensively by Difference Equations giants, Saber Elaydi and Gerry Ladas (and their students and collaborators). In particular we rigorously prove some fascinating conjectures made by Amal Amleh and Gerry Ladas back in 2000. For other conjectures we are content with semi-rigorous proofs. We also extend the work of Emilie Purvine (formerly Hogan) and Doron Zeilberger for rigorously and semi-rigorously proving global asymptotic stability of arbitrary rational difference equations (with positive coefficients).
Added Sept. 12, 2023: Read the interesting message from Evgeni Lozitsky
Added Sept. 19, 2023: Read the equally interesting Second message from Evgeni Lozitsky
Added Oct. 4, 2023: Evgeni Lozitsky wrote it up in a very interesting paper
Added Nov. 23, 2023: Inspired by Evgeni Lozitsky, who used Mathematica, we reproduced and extended his results by writing a Maple package LOZITSKY.txt (see below)
Added Dec. 6, 2023: Evgeni Lozitsky found recurrences of order thirty!
See third message from Evgeni Lozitsky
Added March 20, 2024: Evgeni Lozitsky found many other intriguing conjectures!
See fourth message from Evgeni Lozitsky
x[n+1]=(p+x[n])/x[n-1], where p is positive, and positive initial conditions is ALWAYS bounded, and not only that the quarting equation whose middle roots give you the min and max of the orbit
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x[n+1]=(p+x[n]+x[n-1])/x[n-2], where p is positive, and positive initial conditions is ALWAYS bounded, and not only that the surface in 3D space whose projection to the X-axis gives you the min and max of the orbit
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x[n+1]=(p+x[n]+x[n-1]+...+x[n-k+2])/x[n-k+1],
where p is positive, for k from 1 to 8
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One of us (DZ) offered to donate $100 dollars to the OEIS in honor of Manuel Kauers, if he would meet this challenge, that he brillinatlly met. A donation to the OEIS, in Manuel' honor has been made, as promised.