# A pair of commuting hypergeometric operators on the complex plane and bispectrality

## TL;DR

This paper studies a pair of hypergeometric differential operators on the complex plane, establishing their commutation, spectral decomposition, and connection to a two-dimensional Jacobi transform, revealing new bispectral properties.

## Contribution

It introduces conditions for the commutation of hypergeometric operators and explicitly constructs their joint spectral decomposition using a generalized Jacobi transform.

## Key findings

- Operators commute in the Nelson sense under certain conditions
- Joint spectral decomposition is explicitly constructed
- Inverse transform acts as spectral decomposition for related difference operators

## Abstract

We consider the standard hypergeometric differential operator $D$ regarded as an operator on the complex plane $C$ and the complex conjugate operator $\overline D$. These operators formally commute and are formally adjoint one to another with respect to an appropriate weight. We find conditions when they commute in the Nelson sense and write explicitly their joint spectral decomposition. It is determined by a two-dimensional counterpart of the Jacobi transform (synonyms: generalized Mehler--Fock transform, Olevskii transform). We also show that the inverse transform is an operator of spectral decomposition for a pair of commuting difference operators defined in terms of shifts in imaginary direction.