On a pair of difference equations for the $_4F_3$ type orthogonal polynomials and related exactly-solvable quantum systems

TL;DR

This paper introduces new difference equations for $_4F_3$ type orthogonal polynomials, linking them to exactly-solvable quantum systems, and explores their special cases and applications in quantum models.

Contribution

It presents novel difference equations for $_4F_3$ orthogonal polynomials and demonstrates their use in constructing exactly-solvable quantum models.

Findings

Derived difference equations for $_4F_3$ polynomials.

Linked difference equations to quantum spin chains and fermionic oscillators.

Explored special cases and limit relations for related polynomials.

Abstract

We introduce a pair of novel difference equations, whose solutions are expressed in terms of Racah or Wilson polynomials depending on the nature of the finite-difference step. A number of special cases and limit relations are also examined, which allow to introduce similar difference equations for the orthogonal polynomials of the $ _3 F_2$ and $ _2 F_1$ types. It is shown that the introduced equations allow to construct new models of exactly-solvable quantum dynamical systems, such as spin chains with a nearest-neighbour interaction and fermionic quantum oscillator models.