  The goal of this work is to introduce the notion of $3$-Hom-Lie-dendriform
algebras which is the dendriform version of $3$-Hom-Lie-algebras. They can be
also regarded as the ternary analogous of Hom-Lie-dendriform algebras. We give
the representation of a $3$-Hom-pre-Lie algebra. Moreover, we introduce the
notion of Nijenhuis operators on a $3$-Hom-pre-Lie algebra and provide some
constructions of $3$-Hom-Lie-dendriform algebras in term of Nijenhuis
operators. Parallelly, we introduce the notion of a product and complex
structures on a $3$-Hom-Lie-dendriform algebras and there are also four types
special integrability conditions.
