# §10.44 Sums

## §10.44(i) Multiplication Theorem

 10.44.1 $\mathscr{Z}_{\nu}\left(\lambda z\right)=\lambda^{\pm\nu}\sum_{k=0}^{\infty}% \frac{(\lambda^{2}-1)^{k}(\frac{1}{2}z)^{k}}{k!}\mathscr{Z}_{\nu\pm k}\left(z% \right),$ $|\lambda^{2}-1|<1$. ⓘ Symbols: $!$: factorial (as in $n!$), $\mathscr{Z}_{\NVar{\nu}}\left(\NVar{z}\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter A&S Ref: 9.6.51 Referenced by: §10.44(i), §10.66 Permalink: http://dlmf.nist.gov/10.44.E1 Encodings: TeX, pMML, png See also: Annotations for 10.44(i), 10.44 and 10

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

### Examples

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

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

The restriction $|v|<|u|$ is unnecessary when $\mathscr{Z}=I$ 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 $\left(\tfrac{1}{2}z\right)^{\nu}=\sum_{k=0}^{\infty}(-1)^{k}\frac{(\nu+2k)% \Gamma\left(\nu+k\right)}{k!}I_{\nu+2k}\left(z\right),$ $\nu\neq 0,-1,-2,\ldots$.
 10.44.5 $K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z\right)+\gamma\right)I_{0}% \left(z\right)+2\sum_{k=1}^{\infty}\frac{I_{2k}\left(z\right)}{k},$
 10.44.6 $K_{n}\left(z\right)=\frac{n!(\tfrac{1}{2}z)^{-n}}{2}\sum_{k=0}^{n-1}(-1)^{k}% \frac{(\tfrac{1}{2}z)^{k}I_{k}\left(z\right)}{k!(n-k)}+(-1)^{n-1}\left(\ln% \left(\tfrac{1}{2}z\right)-\psi\left(n+1\right)\right)I_{n}\left(z\right)+(-1)% ^{n}\sum_{k=1}^{\infty}\frac{(n+2k)I_{n+2k}\left(z\right)}{k(n+k)},$

where $\gamma$ is Euler’s constant and $\psi=\ifrac{\Gamma'}{\Gamma}$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).