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• W2963913105 abstract "The Raney numbers are a class of combinatorial numbers generalising the Fuss–Catalan numbers. They are indexed by a pair of positive real numbers (p, r) with p > 1 and 0 < r ⩽ p, and form the moments of a probability density function. For certain (p, r) the latter has the interpretation as the density of squared singular values for certain random matrix ensembles, and in this context equilibrium problems characterising the Raney densities for (p, r) = (θ + 1, 1) and (θ/2 + 1, 1/2) have recently been proposed. Using two different techniques—one based on the Wiener–Hopf method for the solution of integral equations and the other on an analysis of the algebraic equation satisfied by the Green's function—we establish the validity of the equilibrium problems for general θ > 0 and similarly use both methods to identify the equilibrium problem for (p, r) = (θ/q + 1, 1/q), θ > 0 and . The Wiener–Hopf method is used to extend the latter to parameters (p, r) = (θ/q + 1, m + 1/q) for m a non-negative integer, and also to identify the equilibrium problem for a family of densities with moments given by certain binomial coefficients." @default.
• W2963913105 created "2019-07-30" @default.
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• W2963913105 date "2015-06-04" @default.
• W2963913105 modified "2023-10-17" @default.
• W2963913105 title "Equilibrium problems for Raney densities" @default.
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• W2963913105 doi "https://doi.org/10.1088/0951-7715/28/7/2265" @default.
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