Paley–Wiener spaces with vanishing conditions and Painlevé VI transcendents

We modify the classical Paley–Wiener spaces PWx of entire functions of finite exponential type at most x >0, which are square integrable on the real line, via the additional condition of vanishing at finitely many complex points z1, . . . , zn. We compute the reproducing kernels and relate their variations with respect to x to a Krein differential system, whose coefficient (which we call the μ-function) and solutions have determinantal expressions. Arguments specific to the case where the “trivial zeros” z1, . . . , zn are in arithmetic progression on the imaginary axis allow us to establish for expressions arising in the theory a system of two non-linear first order differential equations. A computation, having this non-linear system at his start, obtains quasi-algebraic and among them rational Painlevé transcendents of the sixth kind as certain quotients of such μ-functions. © 2010 Elsevier Inc. All rights reserved