LAUSR

We consider the sequences of matrix bi-orthogonal polynomials with respect to the bilinear forms \(\left\langle {\cdot ,\cdot }\right\rangle _{{{\hat{R}}}}\) and \(\left\langle {\cdot ,\cdot }\right\rangle _{{{\hat{L}}}}\) $$\begin{aligned} \begin{array}{cc} \langle P(z_1),Q(z_2)\rangle _{{\hat{R}}}=\displaystyle \int \limits _{{\mathbb {T}}\times {\mathbb {T}}} P(z_1)^\dag L(z_1)d\mu (z_1,z_2) Q(z_2),&{}\\ &{}\quad P,Q\in {\mathbb {L}}^{p\times p}[z]\\ \left\langle {P(z_1),Q(z_2)}\right\rangle _{{\hat{L}}}=\displaystyle \int \limits _{{\mathbb {T}}\times {\mathbb {T}}} P(z_1)L(z_1)d\mu (z_1,z_2) Q(z_2)^{\dag },&{} \end{array} \end{aligned}$$ where \(\mu (z_1,z_2)\) is a matrix of bi-variate measures supported on \({\mathbb {T}}\times {\mathbb {T}},\) with \({\mathbb {T}}\) the unit circle, \(L^{p\times p}[z]\) is the set of matrix Laurent polynomials of size \(p\times p\) and L(z) is a special polynomial in \(L^{p\times p}[z]\). A connection formula between the sequences of matrix Laurent bi-orthogonal polynomials with respect to \(\left\langle {\cdot , \cdot }\right\rangle _{{{\hat{R}}}}\), (resp. \(\left\langle {\cdot , \cdot }\right\rangle _{{{\hat{L}}}}\)) and the sequence of matrix Laurent bi-orthogonal polynomials with respect to \(d\mu (z_1,z_2)\) is given.