§10.44(i) Multiplication Theorem

 10.44.1 $\mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(\lambda z\right)=\lambda^{\pm\nu}% \sum_{k=0}^{\infty}\frac{(\lambda^{2}-1)^{k}(\frac{1}{2}z)^{k}}{k!}\mathop{% \mathscr{Z}_{\nu\pm k}\/}\nolimits\!\left(z\right),$ $|\lambda^{2}-1|<1$.

If $\mathop{\mathscr{Z}\/}\nolimits=\mathop{I\/}\nolimits$ and the upper signs are taken, then the restriction on $\lambda$ is unnecessary.

Examples

 10.44.2 $\displaystyle\mathop{I_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\mathop{J_{\nu+k}\/}\nolimits% \!\left(z\right),$ $\displaystyle\!\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sum_{k=0}^{\infty}(-1)^{k}\frac{z^{k}}{k!}\mathop{I_{\nu+k}\/}% \nolimits\!\left(z\right).$

 10.44.3 $\mathop{\mathscr{Z}_{\nu}\/}\nolimits\!\left(u\pm v\right)=\sum_{k=-\infty}^{% \infty}(\pm 1)^{k}\mathop{\mathscr{Z}_{\nu+k}\/}\nolimits\!\left(u\right)% \mathop{I_{k}\/}\nolimits\!\left(v\right),$ $|v|<|u|$.

The restriction $|v|<|u|$ is unnecessary when $\mathop{\mathscr{Z}\/}\nolimits=\mathop{I\/}\nolimits$ and $\nu$ is an integer.

Graf’s and Gegenbauer’s Addition Theorems

For results analogous to (10.23.7) and (10.23.8) see Watson (1944, §§11.3 and 11.41).

§10.44(iii) Neumann-Type Expansions

 10.44.4 $\left(\tfrac{1}{2}z\right)^{\nu}=\sum_{k=0}^{\infty}(-1)^{k}\frac{(\nu+2k)% \mathop{\Gamma\/}\nolimits\!\left(\nu+k\right)}{k!}\mathop{I_{\nu+2k}\/}% \nolimits\!\left(z\right),$ $\nu\neq 0,-1,-2,\ldots$.
 10.44.5 $\mathop{K_{0}\/}\nolimits\!\left(z\right)=-\left(\mathop{\ln\/}\nolimits\!% \left(\tfrac{1}{2}z\right)+\EulerConstant\right)\mathop{I_{0}\/}\nolimits\!% \left(z\right)+2\sum_{k=1}^{\infty}\frac{\mathop{I_{2k}\/}\nolimits\!\left(z% \right)}{k},$
 10.44.6 $\mathop{K_{n}\/}\nolimits\!\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{-n}}{2}\sum% _{k=0}^{n-1}(-1)^{k}\frac{(\tfrac{1}{2}z)^{k}\mathop{I_{k}\/}\nolimits\!\left(% z\right)}{k!(n-k)}+(-1)^{n-1}\left(\mathop{\ln\/}\nolimits\!\left(\tfrac{1}{2}% z\right)-\mathop{\psi\/}\nolimits\!\left(n+1\right)\right)\mathop{I_{n}\/}% \nolimits\!\left(z\right)+(-1)^{n}\sum_{k=1}^{\infty}\frac{(n+2k)\mathop{I_{n+% 2k}\/}\nolimits\!\left(z\right)}{k(n+k)},$

where $\EulerConstant$ is Euler’s constant and $\mathop{\psi\/}\nolimits=\ifrac{{\mathop{\Gamma\/}\nolimits^{\prime}}}{\mathop% {\Gamma\/}\nolimits}$5.2).

§10.44(iv) Compendia

For collections of sums and series involving modified Bessel functions see Erdélyi et al. (1953b, §7.15), Hansen (1975), and Prudnikov et al. (1986b, pp. 691–700).