# §10.53 Power Series

 10.53.1 $\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(z\right)=z^{n}\sum_{k=0}^{\infty}% \frac{(-\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!},$
 10.53.2 $\mathop{\mathsf{y}_{n}\/}\nolimits\!\left(z\right)=-\frac{1}{z^{n+1}}\sum_{k=0% }^{n}\frac{(2n-2k-1)!!(\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{n+1}}{z^{n+1}}% \sum_{k=n+1}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.$
 10.53.3 $\displaystyle\mathop{{\mathsf{i}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$ $\displaystyle=z^{n}\sum_{k=0}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1% )!!},$ 10.53.4 $\displaystyle\mathop{{\mathsf{i}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{(-1)^{n}}{z^{n+1}}\sum_{k=0}^{n}\frac{(2n-2k-1)!!(-\frac{1% }{2}z^{2})^{k}}{k!}+\frac{1}{z^{n+1}}\sum_{k=n+1}^{\infty}\frac{(\frac{1}{2}z^% {2})^{k}}{k!(2k-2n-1)!!}.$

For $\mathop{{\mathsf{h}^{(1)}_{n}}\/}\nolimits\!\left(z\right)$ and $\mathop{{\mathsf{h}^{(2)}_{n}}\/}\nolimits\!\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). For $\mathop{\mathsf{k}_{n}\/}\nolimits\!\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).