Let $J_\nu(x):=\displaystyle\sum^\infty_{k=0}\frac{(-1)^k(x/2)^{\nu+2k}}{k!~\Gamma(\nu+k+1)}$ denote a Bessel function. When $\nu\geq0$, let $0<j_{\nu,1}<j_{\nu,2}<\cdots$ denote the positive zeroes of $J_\nu(x)$. My questions are:
$(a)$ Keeping $\nu$ fixed, is it known how the $j_{\nu,k}$'s are distributed on the real line (that is, how fast they increase, whether they accumulate somewhere, etc)?
$(b)$ Also, is there a lower or upper bound on the expression $J_{\nu+1}(j_{\nu,k})$ when $\nu$ is held fixed?
Any help is highly appreciated, thanks!
To be honest, I am absolutely stuck on this. I just know the definition of the Bessel function. I have no clue how to make any conclusions about the zeros of a function from an infinite series. The reason I am interested in this is, the terms $j_{\nu,k}$ and $J_{\nu+1}(j_{\nu,k})$ appear in a formula for an $n$-dimensional Bessel process.
Best Answer
I did some googling, and found that a good $($slightly old$)$ reference for this kind of questions is G.N. Watson's “A treatise on the theory of Bessel functions”. In the $1922$ edition that I currently have access to, the relevant theory appears in Chapter XV. See particularly $15.22$, $15.4$ and $15.81$. This does not answer all the questions, of course, but it's a decent amount of information.
In particular, $15.81$ gives an answer to how fast the zeros of $J_\nu(x)$ grow. $15.22$ tells us that the zeros of $J_\nu$ and $J_{\nu + 1}$ are interlaced. But this does not particularly answer question $(b)$ above.