J. Phys. I France
Volume 2, Numéro 9, September 1992
Page(s) 1721 - 1729
DOI: 10.1051/jp1:1992240
J. Phys. I France 2 (1992) 1721-1729

A non-linear differential equation and a Fredholm determinant

M.L. Mehta

C.E. Saclay, F-91191 Gif-sur-Yvette Cedex, France

(Received 4 March 1992, accepted in final form 5 June 1992)

In several branches of mathematical physics one comes across the Fredholm determinant of the kernel $\sin (x-y)\pi/(x-y)\pi$ on the finite interval ( -t,t). Jimbo, Miwa, Mori and Sato derived a non-linear differential equation for it. We reinvestigate this problem, find five equations satisfied by three functions A(t), B(t) and S(t) related to this Fredholm determinant, and as a consequence deduce the differential equation of Jimbo, Miwa, Mori and Sato.

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