Abstract. We define a novel combinatorial object-the extended Gelfand-Tsetlin graph with cotransition probabilities depending on a parameter q. The boundary of this graph admits an explicit description. We introduce a family of probability measures on the boundary and describe their correlation functions. These measures are a q-analogue of the spectral measures studied earlier in the context of the problem of harmonic analysis on the infinite-dimensional unitary group.Key words: Gelfand-Tsetlin graph, determinantal measures, big q-Jacobi polynomials, basic hypergeometric series.Determinantal measures form a special class of probability measures on spaces of locally finite point configurations (see Borodin's survey [1]). The key property of a determinantal measure is that its correlation functions of any order are expressed in a simple way through a function in two variables, called the correlation kernel.A family of determinantal measures, called the zw-measures, was studied by Borodin and Olshanski [2] in connection with the problem of harmonic analysis on the infinite-dimensional unitary group U (∞), posed by Olshanski [12]. Analogues of the zw-measures also exist for other infinitedimensional classical groups and for infinite-dimensional symmetric spaces (see Olshanski and Osinenko's paper [14]). The zw-measures play a fundamental role in infinite-dimensional harmonic analysis, because their scaling limits govern the spectral decomposition of certain distinguished unitary representations.On the other hand, the zw-measures are nice combinatorial objects and have much in common with the so-called z-measures, which are a particular case of Okounkov's Schur measures on partitions. Like Schur measures, the zw-measures admit a generalization involving an additional deformation parameter similar to Dyson's β-parameter in random matrix theory or the continuous parameter of the Jack symmetric functions (see Olshanski's paper [13]).Our aim is to show that the notion of zw-measures can be extended in another direction; namely, there exists a (nonevident) q-analogue of the zw-measures. Our first result says that the "N -particle q-zw-measures" have a large-N limit, and the limiting probability measure is a determinantal measure on a space of infinite point configurations. We find the corresponding correlation kernelit is expressed through the basic hypergeometric function 2 φ 1 . The large-N limit transition is related to another result, which is of independent interest-a description of the q-boundary for an extended version of the Gelfand-Tsetlin graph.1. The q-analogue of zw-measures. We fix a triple (ζ + , ζ − , q) of real parameters, where ζ + > 0, ζ − < 0, and 0 < q < 1. The corresponding double q-lattice is a subsetBy a configuration we mean an arbitrary subset X ⊂ L. For N = 1, 2, . . . , let G N denote the countable set consisting of all N -point configurations. We enumerate the points of every X ∈ G N in increasing order and write X = (x 1 < · · · < x N ).We are going to introduce, for every N = 1, 2, . . . , a ...
