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.