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This problem is somehow inspired by a bunch of impressive posts of combinatorial identities by T. Amdeberhan. Earlier this month I learnt from computer scientists that they have a generic algorithmic framework of verifying combinatorial identities called Wilf-Zeilberger theory.

In their book [MHD], chapter 9, the authors introduced the operator algebra view of Wilf-Zeilberger theory where they defined hypergeometric sequences (generated by hypergeometric functions) as those eliminated by first-order linear recurrence operators with real polynomial coefficients.

I was wondering if there is any (algebraic) geometric intuition behind their first order Wilf-Zeilberger theory. Loosely speaking, are they trying to figure out some sort of spectrum when conducting the algorithm of verifying combinatoric identities?

The only book I know about operator algebras and geometry is [Moriyoshi & Natsume] whose approach is to do K-theory over $C^*$-algebras. However I do not feel like Wilf-Zeilberger theory is doing that (or I missed something) since their manipulation is over $\mathbb{R}$.

Any reference or comments are welcome!

References

[MHD] Marko Petkovšek, Herbert S. Wilf and Doron Zeilberger. A=B, https://www.math.upenn.edu/~wilf/AeqB.pdf

[Moriyoshi & Natsume] H. Moriyoshi, T. Natsume. Operator algebras and geometry. Vol. 237. American Mathematical Soc., 2008.

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    $\begingroup$ For those who want to know a bit more about operator algebras, there is a nice post mathoverflow.net/questions/200696/… $\endgroup$
    – Henry.L
    Commented Jun 22, 2017 at 17:01
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    $\begingroup$ In the book by Moriyoshi-Natsume, and in the MO question you reference, "operator algebras" are algebras of bounded operators on Hilbert spaces. That doesn't seem to be the same sense in which the term "operator algebra" is used in the book by Petkovšek, Wilf and Zeilberger, since there they are talking about operators on spaces of (possibly) unbounded sequences (e.g. the Fibonacci sequence). On the other hand, as you say, unbounded sequences do arise all the time as spectra of unbounded operators, so it is possible there is more to this than just a coincidence of terminology. $\endgroup$
    – user85913
    Commented Jun 23, 2017 at 8:31
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    $\begingroup$ The algebraic geometry perspective on recurrence relations with polynomial coefficients as used for example in Wilf-Zeilberger theory is $D$-modules. Specifically, such recurrence operators can be transformed into differential operators by transforming $n$ into $x d/dx$ and shift into $x$.Then power series which solve recurrence relations go to formal solutions of differential equations. $\endgroup$
    – Will Sawin
    Commented Nov 17, 2021 at 20:41

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