Dyson's constants in the asymptotics of the determinants of Wiener-Hopf-Hankel operators with the sine kernel

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In this paper we are going to prove two asymptotic formulas for determinants det(I-Kₛ), as s goes to infinity, where Kₛ are the Wiener-Hopf-Hankel operators acting on L²[0,s] with the kernels K(x-y)+K(x+y) and K(x-y)-K(x+y), respectively, and K(t):=sin(t)/(π*t). These formulas were conjectured by Dyson. The identification of the constant term in the asymptotics was an open problem for a long time.