A unified approach to the Dirac fine structures on the $S$-spectrum and a connection with Gegenbauer 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 $(β, m)$, for $β, m \in \mathbb{N}_0$ belong to the kernel of the Dirac-Laplace operators $D^βΔ^m$ of type $(β, m)$ and contain as particular cases axially Poly-Analytic and axially Poly-Harmonic functions. By applying these operators to the Cauchy kernels $S^{-1}_L(s,x)$ of (left) slice hyperholomorphic functions, we obtain an integral representation for axially Poly-Analytic-Harmonic functions. We point out that the kernels $D^βΔ^mS^{-1}_L(s,x)$ have a remarkable connection with Gegenbauer polynomials. By replacing the paravector operator $T$ with commuting components in the kernels $D^βΔ^m S^{-1}_L(s,x)$, we obtain the associated resolvent operators. With these resolvent operators, denoted by $S^{-1}_{L, D^βΔ^m}(s,T)$, we define the associated functional calculi based on the $S$-spectrum and study their properties.