From the hypergeometric differential equation to a non linear Schr\"odinger one

TL;DR

This paper reveals that the q-exponential function satisfies a hypergeometric differential equation, which can be interpreted as a non-linear Schrödinger equation, highlighting connections between special functions and quantum-like equations.

Contribution

It establishes a novel link between hypergeometric functions and non-linear Schrödinger equations, expanding understanding of their mathematical and physical relationships.

Findings

q-exponential function is a hypergeometric function

The hypergeometric differential equation can be viewed as a NLSE

The derived NLSE shares features with the Nobre-Rego Monteiro-Tsallis equation

Abstract

We show that the q-exponential function is a hypergeometric function. Accordingly, it obeys the hypergeometric differential equation. We demonstrate that this differential equation is a non-linear Schr\"odinger equation (NLSE). This NLSE exhibits both similarities and differences vis-a-vis the Nobre-Rego Monteiro-Tsallis one.