In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers \(\left\{ \mathbb {D}_{t}\right\} _{t\in \mathbb {R}}\) are constructed as sub-structures of scaled hypercomplex numbers \(\left\{ \mathbb {H}_{t}\right\} _{t\in \mathbb {R}}\) under the scales (or, the moments) of the set \(\mathbb {R}\) of real numbers. We show that if \(t<0\) , then the classical free probability theory covers our free probability on \(\left\{ \mathbb {D}_{t}\right\} _{t<0}\) ; if \(t>0\) , then our free probability on \(\left\{ \mathbb {D}_{t}\right\} _{t>0}\) is represented by the free probability over the classical hyperbolic numbers \(\mathcal {D}=\mathbb {D}_{1}\) ; and if \(t=0\) , then the free probability on \(\mathbb {D}_{0}\) is actually over the dual numbers \(\textbf{D}=\mathbb {D}_{0}\) . Since the usual free probability theory is over \(\mathbb {C}\) , we here concentrate on establishing our free probability theory on \(\mathcal {D}\) , or that on \(\textbf{D}\) . Our approaches are motivated by the Speicher’s combinatorial free probability. As applications, the \(\mathbb {D}_{t}\) -free-probabilistic versions of semicircular elements and circular elements are considered.
