A multi-dimensional Markov chain and the Meixner ensemble

We show that the transition probability of the Markov chain (G(i,1),...,G(i,n)) (ia parts per thousand yen1), 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.