Special functions from quantum canonical transformations

TL;DR

This paper uses quantum canonical transformations to derive integral representations and solutions for hypergeometric equations and the non-periodic Toda equation, revealing a new generalized hypergeometric framework.

Contribution

It introduces a novel approach linking quantum canonical transformations to hypergeometric functions and Toda equations, expanding the analytical tools available for these special functions.

Findings

Derived integral representations for hypergeometric solutions

Obtained solutions for the non-periodic Toda equation

Proposed a two-dimensional generalized hypergeometric equation

Abstract

Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body Toda equation are also found. The derivation of these representations motivate the form of a two-dimensional generalized hypergeometric equation which contains the non-periodic Toda equation as a special case and whose solutions may be obtained by quantum canonical transformation.