The relation (A.42) can be verified by the power series definition of $I_{\nu}$ and $\Ku(\nu+\frac{1}{2},2\nu+1,2y)$, so we just omit the computations.

To prove (A.43), first we assume $\nu$ is not an integer. Combining the definition

(A.44)

$$K_{\nu}(y)=\frac{\pi}{\sin(\nu\pi)}\cdot\frac{I_{-\nu}(y)-I_{\nu}(y)}{2}$$

and the relation

(A.45)

$$\mathcal{U}(\nu+\frac{1}{2},2\nu+1,y)=\frac{\Gamma(-2\nu)}{\Gamma(\frac{1}{2}-\nu)}\cdot\Ku(\nu+\frac{1}{2},2\nu+1,y)+\frac{\Gamma(2\nu)}{\Gamma(\nu+\frac{1}{2})}\cdot y^{-2\nu}\cdot\Ku(\frac{1}{2}-\nu,1-2\nu,y),$$

which is given by (A.22). If $\nu$ is an integer, the relation (A.43) can be obtained by the limiting definition of $K_{\nu}$ and the continuity argument for $\nu$.

∎