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# Explicit formulas of the heat kernel on the quaternionic projective spaces

## Formules explicites du noyau de la chaleur sur les espaces projectifs des quaternions

Ali Hafoud
Centre Régional des Métiers de l’Education et de la Formation de Kenitra
Maroc

Allal Ghanmi
Université Mohammed-V de Rabat
Maroc

Published on 14 March 2022   DOI : 10.21494/ISTE.OP.2022.0809

### Mots-clés

We consider the heat equation on the quaternionic projective space $P_{n}(ℍ)$, and we establish two formulas for the heat kernel, a series expansion involving Jacobi polynomials, and an integral representation involving a $\theta$-function. More precisely, using the quaternonic Hopf fibration and the explicit integral representation of the heat kernel on the complex projective space $P_{2n+1}(ℂ)$, as well as an integral representation of Jacobi polynomials in terms of Gegenbauer polynomials, we give an explicit integral representation of the heat kernel on the $n$-quaternionic projective space. We also establish an explicit series expansion of the heat kernel in terms of the Jacobi polynomials. Moreover, we derive an explicit formula of the heat kernel on the sphere $S^4$.

We consider the heat equation on the quaternionic projective space $P_{n}(ℍ)$, and we establish two formulas for the heat kernel, a series expansion involving Jacobi polynomials, and an integral representation involving a $\theta$-function. More precisely, using the quaternonic Hopf fibration and the explicit integral representation of the heat kernel on the complex projective space $P_{2n+1}(ℂ)$, as well as an integral representation of Jacobi polynomials in terms of Gegenbauer polynomials, we give an explicit integral representation of the heat kernel on the $n$-quaternionic projective space. We also establish an explicit series expansion of the heat kernel in terms of the Jacobi polynomials. Moreover, we derive an explicit formula of the heat kernel on the sphere $S^4$.