Realization of Bessel functions

Question

As I know when whe trying to execute Bessel functions when $z < 8$ we using this formula $$J_\nu(z) = (\frac{1}{2}z)^\nu \sum_{k = 0}^{\infty} \frac{(-\frac{1}{4}z^2)^k}{k!\Gamma(\nu+k+1)}$$ And thats formula I found when $z \geq 8$ $$J_\nu(z) \simeq \sqrt{\frac{2}{\pi z}}[\cos(z - \frac{\nu \pi}{2} - \frac{\pi}{4})P_\nu(z) - \sin(z - \frac{\nu \pi}{2} - \frac{\pi}{4})Q_v(z)]$$ So, please, could you help me to understand what is and how to execute $P_\nu(z)$ and $Q_v(z)$ here?

Answer 1

Oh, yep, i found it! It's about Hankel asymptotic expansions and looks like $$ P_\nu(z) \equiv 1 - \frac{(\nu,2)}{(2z)^2} + \frac{(\nu,4)}{(2z)^4} - \dots $$ $$ Q_\nu(z) \equiv \frac{(\nu,1)}{(2z)} - \frac{(\nu,3)}{(2z)^3} + \frac{(\nu,5)}{(2z)^5} - \dots $$ where $$ (\nu,m) = \frac{(4\nu^2 - 1^2)(4\nu^2 - 3^2) \dots (4\nu^2 - (2m -1)^2)}{2^{2m} m!}, m = 1,2,3,\cdots $$