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With a given holomorphic section of a Hermitian vector bundle, one can associate a residue current by means of Cauchy–Fantappiè–Leray type formulas. In this paper we define products of such residue currents. We prove that, in the case of a complete intersection, the product of the residue currents of a tuple of sections coincides with the residue current of the direct sum of the sections.

We show that the transition probability of the Markov chain ($G$($i$,1),...,$G$($i$,$n$))_{$i$≥1}, where the$G$($i$,$j$)’s are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for$G$($m$,$n$). We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.

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