  This paper addresses a Hom-associative algebra built as a direct sum of a
given Hom-associative algebra $(\mathcal{A}, \cdot, \alpha)$ and its dual
$(\mathcal{A}^{\ast}, \circ, \alpha^{\ast}),$ endowed with a non-degenerate
symmetric bilinear form $\mathcal{B},$ where $\cdot$ and $\circ$ are the
products defined on $\mathcal{A}$ and $\mathcal{A}^{\ast},$ respectively, and $
\alpha$ and $\alpha^{\ast}$ stand for the corresponding algebra homomorphisms.
Such a double construction, also called Hom-Frobenius algebra, is interpreted
in terms of an infinitesimal Hom-bialgebra. The same procedure is applied to
characterize the double construction of biHom-associative algebras, also called
biHom-Frobenius algebra. Finally, a double construction of Hom-dendriform
algebras, also called double construction of Connes cocycle or symplectic
Hom-associative algebra, is performed. Besides, the concept of biHom-dendriform
algebras is introduced and discussed. Their bimodules and matched pairs are
also constructed, and related relevant properties are given.
