# §10.8 Power Series

For $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$ see (10.2.2) and (10.4.1). When $\nu$ is not an integer the corresponding expansions for $\mathop{Y_{\nu}\/}\nolimits\!\left(z\right)$, $\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)$, and $\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)$ are obtained by combining (10.2.2) with (10.2.3), (10.4.7), and (10.4.8).

When $n=0,1,2,\ldots$,

 10.8.1 $\mathop{Y_{n}\/}\nolimits\!\left(z\right)=-\frac{(\tfrac{1}{2}z)^{-n}}{\pi}% \sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\left(\tfrac{1}{4}z^{2}\right)^{k}+\frac{2}% {\pi}\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}z\right)\mathop{J_{n}\/}% \nolimits\!\left(z\right)-\frac{(\tfrac{1}{2}z)^{n}}{\pi}\sum_{k=0}^{\infty}(% \mathop{\psi\/}\nolimits\!\left(k+1\right)+\mathop{\psi\/}\nolimits\!\left(n+k% +1\right))\frac{(-\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!},$

where $\mathop{\psi\/}\nolimits\!\left(x\right)=\mathop{\Gamma\/}\nolimits'\!\left(x% \right)/\mathop{\Gamma\/}\nolimits\!\left(x\right)$5.2(i)). In particular,

 10.8.2 $\mathop{Y_{0}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\left(\mathop{\ln\/}% \nolimits\!\left(\tfrac{1}{2}z\right)+\gamma\right)\mathop{J_{0}\/}\nolimits\!% \left(z\right)+\frac{2}{\pi}\left(\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}-(1+\tfrac% {1}{2})\frac{(\tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})% \frac{(\tfrac{1}{4}z^{2})^{3}}{(3!)^{2}}-\cdots\right),$

where $\gamma$ is Euler’s constant (§5.2(ii)).

For negative values of $n$ use (10.4.1).

The corresponding results for $\mathop{{H^{(1)}_{n}}\/}\nolimits\!\left(z\right)$ and $\mathop{{H^{(2)}_{n}}\/}\nolimits\!\left(z\right)$ are obtained via (10.4.3) with $\nu=n$.

 10.8.3 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{J_{\mu}\/}\nolimits\!\left(% z\right)=(\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(-% \tfrac{1}{4}z^{2})^{k}}{k!\mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)% \mathop{\Gamma\/}\nolimits\!\left(\mu+k+1\right)}.$