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We found that when the spinless model is off the half-filling regime ($\mu\neq V$), the Helmholtz free energy (HFE) can be written as two$\beta$-expansions: one expansion comes from the half-filling configuration andanother one that depends on the parameter $x = \mu - V$. We show numericallythat the chemical potential as a function of temperature satisfies a relationsimilar to the one derived from the particle-hole symmetry of the fermionicspinless model. We extend the $\beta$-expansion of the HFE of theone-dimensional fermionic spinless Hubbard model up to order $\beta^8$.

The $\beta$-expansion of the $D=1$ fermionic spinless Hubbard model off the half-filling regime