We present for the first time the asymptotic expressions of the one-body distribution functions of singular values for the Laguerre ensemble and the fixed-trace ensemble in a systematic and unified way. These expressions can be utilized in a wide range of research fields such as data mining, time series analysis and quantum information where finite-size effects play a crucial role. The asymptotic analysis is performed based on our finding of systems of partial differential equations for the confluent Selberg–Kaneko integral and the simplex-type Selberg–Kaneko integral, both of which are introduced in Kubotani et al (2008 Phys. Rev. Lett. 100 240501) and Adachi et al (2009 Ann. Phys., NY 324 2278). Thus, we demonstrate that the systems provide us with new resources of special functions with multiple variables, which can be valuable for analytic studies in various areas.
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