The modified Bessel Function of the Second Kind $K_v(z)$ can be expressed by $$ K_v(z)=\frac{\pi\csc(\pi v)}{2}(I_{-v}(z)-I_v(z)), $$ if $v$ is not an integer. $K_v(z)$ satisfies the following multiplication theorem. $$ \lambda^{v}K_v(\lambda z)=\sum_{l=0}^{\infty}{\frac{(-1)^l}{l!}(\frac{(\lambda^2-1)z}{2})^jK_{v-j}(z)}, ~~\text{if}~~ |\lambda^2-1|<1. $$ My question is that whether the condition $|\lambda^2-1|<1$ here is necessary at least when $\lambda$ is real.
I saw this formula in the book 'Handbook of Mathematical Functions' edited by Abramowitz and Stegun. However, when I checked in Wikipedia, it seems that this condition is not necessary and the formula is valid for any $\lambda$. Can anyone show me which is correct?