# §10.66 Expansions in Series of Bessel Functions

 10.66.1 $\mathop{\mathrm{ber}_{\nu}\/}\nolimits x+i\mathop{\mathrm{bei}_{\nu}\/}% \nolimits x=\sum_{k=0}^{\infty}\frac{e^{(3\nu+k)\pi i/4}x^{k}\mathop{J_{\nu+k}% \/}\nolimits\!\left(x\right)}{2^{k/2}k!}=\sum_{k=0}^{\infty}\frac{e^{(3\nu+3k)% \pi i/4}x^{k}\mathop{I_{\nu+k}\/}\nolimits\!\left(x\right)}{2^{k/2}k!}.$
 10.66.2 $\displaystyle\mathop{\mathrm{ber}_{n}\/}\nolimits\!\left(x\sqrt{2}\right)$ $\displaystyle=\sum_{k=-\infty}^{\infty}(-1)^{n+k}\mathop{J_{n+2k}\/}\nolimits% \!\left(x\right)\mathop{I_{2k}\/}\nolimits\!\left(x\right),$ $\displaystyle\mathop{\mathrm{bei}_{n}\/}\nolimits\!\left(x\sqrt{2}\right)$ $\displaystyle=\sum_{k=-\infty}^{\infty}(-1)^{n+k}\mathop{J_{n+2k+1}\/}% \nolimits\!\left(x\right)\mathop{I_{2k+1}\/}\nolimits\!\left(x\right).$