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We consider mixed type multiple orthogonal polynomials associated with asystem of weight functions consisting of two vectors. One vector is defined interms of scaled modified Bessel function of the first kind $I_\mu$ and$I_{\mu+1}$, the other vector is defined in terms of scaled modified Besselfunction of the second kind $K_\nu$ and $K_{\nu+1}$. We show that thecorresponding mixed type multiple orthogonal polynomials exist. For the specialcase that each multi-index is on or close to the diagonal, basic properties ofthe polynomials and their linear forms are investigated, which include explicitformulas, integral representations, differential properties, limiting forms andrecurrence relations. It comes out that, for specified parameters, the linearforms of these mixed type multiple orthogonal polynomials can be interpreted asbiorthogonal functions encountering in recent studies of products of twocoupled random matrices. This particularly implies a Riemann-Hilbertcharacterization of the correlation kernel, which provides an alternative wayfor further asymptotic analysis.

Mixed type multiple orthogonal polynomials associated with the modified Bessel functions and products of two coupled random matrices