Matrix-valued <i>θ</i>-deformed bi-orthogonal polynomials, non-commutative Toda theory and Bäcklund transformation

Abstract 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 τ -functions. At the same time, Bäcklund transformations are realised 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.