Connection formulae for Painlevé V functions. II: the \(\delta\) function Bose gas problem (Q1109986)

[For part I see ibid. 19, 42-72 (1986; Zbl 0639.58041).] We continue our study of those Painlevé V equations which admit a one- parameter family of solutions analytic at the origin and solve the global problem of connecting the 2-parameter expansion about \(t=0\) with the 2- parameter expansion valid for \(t\to \pm i\infty\). A special case of our results is applied to the study of the one-particle reduced density matrix of the one-dimensional gas of impenetrable bosons. We also use our results to obtain a 2-parameter connection formula for a class of Painlevé III functions.