# Differential operators, radial parts and a one-parameter family of hypergeometric functions of type BC

## TL;DR

This paper introduces a new family of hypergeometric functions of type BC, exploring their properties as eigenfunctions of differential operators, extending the Heckman-Opdam framework with a focus on positivity and asymptotics.

## Contribution

It defines symmetric and non-symmetric hypergeometric functions of type BC as eigenfunctions of differential and differential-reflection operators, expanding the theory with new parameter conditions.

## Key findings

- Established positivity and boundedness properties.
- Derived asymptotic estimates.
- Extended Heckman-Opdam hypergeometric functions to type BC.

## Abstract

We introduce the symmetric (respectively, non-symmetric) $\tau_{-\ell}-$hypergeometric functions associated with a root system of type $BC$ as joint eigenfunctions of a commutative algebra of differential (respectively, differential-reflection) operators. Under certain conditions on the real parameter $\ell$, we derive their properties (positivity, estimates, asymptotics and boundedness) by establishing the analogous properties for the Heckman-Opdam (symmetric and non-symmetric) hypergeometric functions corresponding to (not necessarily positive) multiplicity functions which are standard.