Special non uniform lattice ($snul$) orthogonal polynomials on discrete dense sets of points.

TL;DR

This paper introduces special non-uniform lattices in difference calculus that support orthogonal polynomials satisfying difference relations, expanding the understanding of polynomial behavior on dense $q$-calculus lattices.

Contribution

It presents new orthogonal polynomials defined on non-uniform lattices compatible with difference calculus, especially on dense $q$-lattices, which was not previously explored.

Findings

Orthogonal polynomials on special non-uniform lattices are constructed.

Difference relations are established for these polynomials on dense $q$-lattices.

The work extends the theory of polynomials in $q$-calculus to new lattice structures.

Abstract

Difference calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree) can only be made on special lattices well known in contemporary $q-$calculus. Orthogonal polynomials satisfying difference relations on such lattices are presented. In particular, lattices which are dense on intervals ($|q|=1$) are considered.