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4: 10.29 Recurrence Relations and Derivatives
§10.29(i) Recurrence Relations
With $\mathscr{Z}_{\nu}\left(z\right)$ defined as in §10.25(ii), …
$I_{0}'\left(z\right)=I_{1}\left(z\right),$
For results on modified quotients of the form $\ifrac{z\mathscr{Z}_{\nu\pm 1}\left(z\right)}{\mathscr{Z}_{\nu}\left(z\right)}$ see Onoe (1955) and Onoe (1956).
§10.29(ii) Derivatives
They are analogous to the addition theorems for Bessel functions (§10.23(ii)) and modified Bessel functions (§10.44(ii)). …
6: 10.28 Wronskians and Cross-Products
§10.28 Wronskians and Cross-Products
10.28.1 $\mathscr{W}\left\{I_{\nu}\left(z\right),I_{-\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)I_{-\nu-1}\left(z\right)-I_{\nu+1}\left(z\right)I_{-\nu}\left(z% \right)=-2\sin\left(\nu\pi\right)/(\pi z),$
10.28.2 $\mathscr{W}\left\{K_{\nu}\left(z\right),I_{\nu}\left(z\right)\right\}=I_{\nu}% \left(z\right)K_{\nu+1}\left(z\right)+I_{\nu+1}\left(z\right)K_{\nu}\left(z% \right)=1/z.$
8: 28.1 Special Notation
and the modified Mathieu functions
 $\operatorname{Ce}_{\nu}\left(z,q\right)$, $\operatorname{Se}_{\nu}\left(z,q\right)$, $\operatorname{Fe}_{n}\left(z,q\right)$, $\operatorname{Ge}_{n}\left(z,q\right)$, $\operatorname{Me}_{\nu}\left(z,q\right)$, ${\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)$, ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)$, ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)$, …
The functions ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)$ are also known as the radial Mathieu functions. … The radial functions ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)$ are denoted by ${\operatorname{Mc}^{(j)}_{n}}\left(z,q\right)$ and ${\operatorname{Ms}^{(j)}_{n}}\left(z,q\right)$, respectively.
9: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
and $\widetilde{I}_{\nu}\left(x\right)$, $\widetilde{K}_{\nu}\left(x\right)$ are real and linearly independent solutions of (10.45.1): … The corresponding result for $\widetilde{K}_{\nu}\left(x\right)$ is given by …
10: 10.35 Generating Function and Associated Series
§10.35 Generating Function and Associated Series
10.35.4 $1=I_{0}\left(z\right)-2I_{2}\left(z\right)+2I_{4}\left(z\right)-2I_{6}\left(z% \right)+\dotsb,$
10.35.5 $e^{\pm z}=I_{0}\left(z\right)\pm 2I_{1}\left(z\right)+2I_{2}\left(z\right)\pm 2% I_{3}\left(z\right)+\dotsb,$
$\cosh z=I_{0}\left(z\right)+2I_{2}\left(z\right)+2I_{4}\left(z\right)+2I_{6}% \left(z\right)+\dots,$
$\sinh z=2I_{1}\left(z\right)+2I_{3}\left(z\right)+2I_{5}\left(z\right)+\dots.$