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Summary:
The Riccati equations as well as some interesting inequalities for the ratios of Bessel functions of purely imaginary argument $T_p(x;1)=\frac {K_p(x)}{xK_{p+1}(x)}$ and $T_p(x;-1)=\frac {I_p(x)}{xI_{p-1}(x)}$ are derived. Solutions of the Riccati equations are given in terms of power series in $p^{-1}$. In particular, the asymptotic formulae for $T_p(x;\pm 1)$ with a remainder of order $O(p^{-5})$ are obtained. For $p$ small, they represent an asymptotic expansion in $x$ up to the order $O(x^{-5})$.
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