# Series that looks like the modified Bessel function

The modified Bessel function is defined as: $$I_{\nu}(z)=\sum_{k=0}^{\infty}\frac{(z/2)^{2k+\nu}}{k!\hspace{0.05cm}\Gamma(k+\nu+1)}$$

However I have the following series $$\sum_{k=0}^{\infty}\frac{k\hspace{0.05cm}(z/2)^{2k+\nu}}{k!\hspace{0.05cm}\Gamma(k+\nu+1)}$$

Any idea what it is or how is it related to the $I_{\nu}(z)$?

One has, by a change of index, $p:=k-1$, $k=p+1$, \begin{align} \sum_{k=0}^{\infty}\frac{k\hspace{0.05cm}(z/2)^{2k+\nu}}{k!\hspace{0.05cm}\Gamma(k+\nu+1)}&=\color{red}{0}+\sum_{k=1}^{\infty}\frac{k\:(z/2)^{2k+\nu}}{k!\hspace{0.05cm}\Gamma(k+\nu+1)} \\\\&=\sum_{k=1}^{\infty}\frac{(z/2)^{2k+\nu}}{(k-1)!\hspace{0.05cm}\Gamma(k+\nu+1)} \\\\&=\sum_{p=0}^{\infty}\frac{(z/2)^{2p+2+\nu}}{p!\hspace{0.05cm}\Gamma(p+\nu+2)} \\\\&=\frac{z}2 \cdot I_{\nu+1}(z). \end{align}