Connection problem for special function solutions of Painlevé III equation

In this paper we compute the small and large \(x\) asymptotics of the special function solutions of Painlevé-III equation. We use the representation of solution in terms of Toeplitz determinants of Bessel functions, which is new. Toeplitz determinants are rewritten as multiple contour integrals using Andrèief identity. Finally small and large \(x\) asymptotics is obtained using elementary asymptotic methods. The claimed result has not appeared in the literature before. Our formulas are useful for numerical computations of corresponding solutions of Painlevé-III equation.