I'm interested in the the behaviour of product
$$Y_m(z)J_m(z)$$
as a function of $m \in \mathbb{R}_+$, for fixed, real $z > 0$?
Here, $ J_m(z)$ and $Y_m(z)$ are Bessel functions of the first and second kind and order $m$, respectively.
Numerical computation suggests it goes to zero as $m \to \infty$, and the asymptotic expansions in Abramowitz and Stegun (p. 365) confirms this. However, I am more interested in the behaviour in the non-asymptotic situation.
There is a huge amount of literature on Bessel functions, but I somehow cannot seem to find anything concerning the behaviour of the particular product above. All help is much appreciated.