Singularity confinement for a class of m-th order difference equations of combinatorics
(2008) Royal Society of London. Philosophical Transactions. Mathematical, Physical and Engineering Sciences — Vol. 366, n° 1867, p. 877-922 (2008)
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- Abstract
- In a recent publication, it was shown that a large class of integrals over the unitary group U(n) satisfy nonlinear, non-autonomous difference equations over n, involving a finite number of steps; special cases are generating functions appearing in questions of the longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the discrete Painleve property; roughly speaking, this means that after a finite number of steps the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be. finite again ('singularity confinement'). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the Painleve property for the discrete relations is inherited from the Painleve property of the (continuous) Toeplitz lattice.
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Adler, M., Van Moerbeke, P., & Vanhaecke, P. (2008). Singularity confinement for a class of m-th order difference equations of combinatorics. Royal Society of London. Philosophical Transactions. Mathematical, Physical and Engineering Sciences, 366(1867), 877-922. https://doi.org/10.1098/rsta.2007.2090 (Original work published 2008)