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.