Translation Operator and Heat Equation Analysis for the Generalized Linear Canonical Fourier-Bessel Transform

Résumé

In this work, we introduce a new translation operator naturally associated with the generalized linear canonical Fourier--Bessel transform \( \mathcal{F}^{m}_{\alpha,n} \). This operator is constructed through an appropriate Cauchy problem involving a generalized Bessel-type differential operator and extends several known translation structures in harmonic analysis. We establish its main analytical properties and use it to define a convolution structure adapted to the generalized linear canonical Fourier--Bessel framework. Furthermore, we apply this convolution approach to the study of the heat equation governed by the conjugate of the generalized Bessel-type operator \(\Delta^{m}_{\alpha,n} \). An explicit representation of the solution is obtained via a generalized heat kernel, highlighting the effectiveness of the proposed method and its potential applications.