# §10.50 Wronskians and Cross-Products

 10.50.1 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{\mathsf{j}_{n}\/}% \nolimits\!\left(z\right),\mathop{\mathsf{y}_{n}\/}\nolimits\!\left(z\right)\right\}$ $\displaystyle=z^{-2},$ $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{h}^{(1)}_{% n}}\/}\nolimits\!\left(z\right),\mathop{{\mathsf{h}^{(2)}_{n}}\/}\nolimits\!% \left(z\right)\right\}$ $\displaystyle=-2iz^{-2}.$
 10.50.2 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{i}^{(1)}_{% n}}\/}\nolimits\!\left(z\right),\mathop{{\mathsf{i}^{(2)}_{n}}\/}\nolimits\!% \left(z\right)\right\}$ $\displaystyle=(-1)^{n+1}z^{-2},$ $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{i}^{(1)}_{% n}}\/}\nolimits\!\left(z\right),\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(z% \right)\right\}$ $\displaystyle=\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathsf{i}^{(2)}_% {n}}\/}\nolimits\!\left(z\right),\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(z% \right)\right\}\\ =-\tfrac{1}{2}\pi z^{-2}.$
 10.50.3 $\displaystyle\mathop{\mathsf{j}_{n+1}\/}\nolimits\!\left(z\right)\mathop{% \mathsf{y}_{n}\/}\nolimits\!\left(z\right)-\mathop{\mathsf{j}_{n}\/}\nolimits% \!\left(z\right)\mathop{\mathsf{y}_{n+1}\/}\nolimits\!\left(z\right)$ $\displaystyle=z^{-2},$ $\displaystyle\mathop{\mathsf{j}_{n+2}\/}\nolimits\!\left(z\right)\mathop{% \mathsf{y}_{n}\/}\nolimits\!\left(z\right)-\mathop{\mathsf{j}_{n}\/}\nolimits% \!\left(z\right)\mathop{\mathsf{y}_{n+2}\/}\nolimits\!\left(z\right)$ $\displaystyle=(2n+3)z^{-3}.$
 10.50.4 $\mathop{\mathsf{j}_{0}\/}\nolimits\!\left(z\right)\mathop{\mathsf{j}_{n}\/}% \nolimits\!\left(z\right)+\mathop{\mathsf{y}_{0}\/}\nolimits\!\left(z\right)% \mathop{\mathsf{y}_{n}\/}\nolimits\!\left(z\right)=\mathop{\cos\/}\nolimits\!% \left(\tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor n/2\right\rfloor}(-1)^{k% }\frac{a_{2k}(n+\tfrac{1}{2})}{z^{2k+2}}+\mathop{\sin\/}\nolimits\!\left(% \tfrac{1}{2}n\pi\right)\sum_{k=0}^{\left\lfloor(n-1)/2\right\rfloor}(-1)^{k}% \frac{a_{2k+1}(n+\tfrac{1}{2})}{z^{2k+3}},$

where $a_{k}(n+\tfrac{1}{2})$ is given by (10.49.1).

Results corresponding to (10.50.3) and (10.50.4) for $\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$ and $\mathop{{\mathsf{i}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$ are obtainable via (10.47.12).