A unified approach to the Dirac fine structures on the $S$-spectrum and a connection with Jacobi polynomials

This paper contributes to the recently introduced theory of fine structures on the $S$-spectrum. We study, in a unified way, the functional calculi for axially Poly-Analytic-Harmonic functions on the $S$-spectrum. Axially Poly-Analytic-Harmonic functions of type $(\beta ,m)$, for $\beta ,m\in {\mathbb{N}}_0$ belong to the kernel of the Dirac-Laplace operators $D^{\beta }{\mathrm{\Delta }}_{n+1}^m$ of type $(\beta ,m)$ and contain as particular cases Poly-Analytic and Poly-Harmonic functions of axial type. By applying these operators to the Cauchy kernels $S_L^{-1}(s,x)$ of (left) slice hyperholomorphic functions, we obtain an integral representation for axially Poly-Analytic-Harmonic functions. We point out that the kernels $D^{\beta }{\mathrm{\Delta }}_{n+1}^m{S}_L^{-1}(s,x)$ have a remarkable connection with Jacobi polynomials. By replacing the paravector operator $T$ with commuting components in the kernels $D^{\beta }{\mathrm{\Delta }}_{n+1}^m{S}_L^{-1}(s,x)$, we obtain the associated resolvent operators. With these resolvent operators, denoted by $S_{L,D^{\beta }{\mathrm{\Delta }}^m}^{-1}(s,T)$, we define the associated functional calculi based on the $S$-spectrum and study their properties.