This paper is devoted to revealing the relationship between matrix-valued
θ-deformed bi-orthogonal polynomials and non-commutative Toda-type
hierarchies. In this procedure, Wronski quasi-determinants are widely used and
play the role of non-commutative Ï„-functions. At the same time, B\"acklund
transformations are realized by using a moment modification method and
non-commutative θ-deformed Volterra hierarchies are obtained, which
contain the known examples of the Itoh-Narita-Bogoyavlensky lattices and the
fractional Volterra hierarchy.Comment: 30 pages. Comments are welcom