Eigenfunctions of $GL(N,\text{RR})$ Toda chain: The Mellin-Barnes representation

The recurrent relations between the eigenfunctions for $GL(N,\text{RR})$ and $GL(N-1,\text{RR})$ quantum Toda chains is derived. As a corollary, the Mellin-Barnes integral representation for the eigenfunctions of a quantum open Toda chain is constructed for the $N$-particle case.