New sharp bounds for the Jacobi heat kernel via an extension of the Dijksma-Koornwinder formula

TL;DR

This paper derives precise bounds for the Jacobi heat kernel in new parameter ranges, extending previous results by generalizing the Dijksma-Koornwinder formula for Jacobi polynomials.

Contribution

The authors extend the Dijksma-Koornwinder formula to obtain sharp heat kernel bounds for Jacobi polynomials in previously unaddressed parameter ranges.

Findings

Established sharp bounds for Jacobi heat kernel in new parameter regimes

Generalized the Dijksma-Koornwinder product formula

Completed the authors' earlier partial results

Abstract

We obtain sharp estimates for the Jacobi heat kernel in a range of parameters where the result has not been established before. This extends and completes an earlier result due to the authors. The proof is based on a generalization of the Dijksma-Koornwinder product formula for Jacobi polynomials.