Orthogonal Polynomials appearing in SIE grid representations

TL;DR

This paper explores how orthogonal polynomials naturally emerge in specific algebraic representations of grid-shaped quivers, linking algebraic structures to classical special functions.

Contribution

It introduces the concept of SIE quiver representations and demonstrates their connection to orthogonal polynomials like Laguerre and Legendre-Gegenbauer.

Findings

Orthogonal polynomials appear in SIE quiver representations.

Ladder commutator conditions are key to the SIE property.

Framework unifies algebraic and analytical aspects of special functions.

Abstract

We show in this article how orthogonal polynomials appear in certain representations of grid shaped quivers. After a short introduction into the general notion of quivers and their representations by linear operators we define the notion of an SIE quiver representation: All intrinsic endomorphisms arising from circuits in the quiver act as scalar multipliers. We then present several lemmas that ensure this SIE property of a quiver representation. Ladder commutator conditions and certain diagram commutativity "up to scalar multiples" play a central role. The theory will then be applied to three examples. Extensive calculations shows how Associated Laguerre, Legendre--Gegenbauer polynomials and binomial distributions fit into the framework of grid shaped SIE quivers. One can see, that this algebraic point of view is foundational for orthogonal polynomials and special functions.