# §10.31 Power Series

For $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)$ see (10.25.2) and (10.27.1). When $\nu$ is not an integer the corresponding expansion for $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)$ is obtained from (10.25.2) and (10.27.4).

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

 10.31.1 $\mathop{K_{n}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{-n}\sum% _{k=0}^{n-1}\frac{(n-k-1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\mathop{\ln% \/}\nolimits\!\left(\tfrac{1}{2}z\right)\mathop{I_{n}\/}\nolimits\!\left(z% \right)+(-1)^{n}\tfrac{1}{2}(\tfrac{1}{2}z)^{n}\sum_{k=0}^{\infty}\left(% \mathop{\psi\/}\nolimits\!\left(k+1\right)+\mathop{\psi\/}\nolimits\!\left(n+k% +1\right)\right)\frac{(\tfrac{1}{4}z^{2})^{k}}{k!(n+k)!},$

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

 10.31.2 $\mathop{K_{0}\/}\nolimits\!\left(z\right)=-\left(\mathop{\ln\/}\nolimits\!% \left(\tfrac{1}{2}z\right)+\EulerConstant\right)\mathop{I_{0}\/}\nolimits\!% \left(z\right)+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1+\tfrac{1}{2})\frac{(% \tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1% }{4}z^{2})^{3}}{(3!)^{2}}+\cdots.$

For negative values of $n$ use (10.27.3).

 10.31.3 $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)\mathop{I_{\mu}\/}\nolimits\!\left(% z\right)=(\tfrac{1}{2}z)^{\nu+\mu}\sum_{k=0}^{\infty}\frac{(\nu+\mu+k+1)_{k}(% \tfrac{1}{4}z^{2})^{k}}{k!\mathop{\Gamma\/}\nolimits\!\left(\nu+k+1\right)% \mathop{\Gamma\/}\nolimits\!\left(\mu+k+1\right)}.$