A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants

We consider Fredholm determinants of matrix Hankel operators associated to matrix versions of the $n$-th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix-valued version of the Lenard operators. In particular, the Riemann–Hilbert techniques used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitly written in terms of the matrix-valued Lenard operators and some solutions of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy Hankel operators.