Let $j_{m,n}$ be the $n^{th}$ zero of the $m^{th}$ order Bessel function $J_{m}(x)$. I would like to know the asymptotic behaviour of the first zeros of the Bessel function $j_{m,1}$, as the order $m$ of the Bessel function tends to infinity.
Numerically it seems to go very roughly as $j_{m,1}\sim m$, but I am interested in the specific behaviour and this is something I'm struggling to find, or work out, myself.
I am also interested in the same behaviour for the zeros $j_{m,n}'$ of the derivative of the $m^{th}$ order Bessel function, $J_{m}'(x)$.
Thanks in advance for any help.
Hoping that my memory is still correct, I think that they are given by something like $$j_{m,1}=m+\sum_{k=0}^p a_k \,m^{\frac{1-k}3}$$ but I do not think that the coefficients are known. Tricomi proposed in $1949$ $$j_{m,k}=m+a_k\, m^{\frac 13}+O\left(m^{-\frac 13}\right)$$
I generated a table for $m$ from $10$ to $1000$ by steps of $10$ and made a curve fit. Its results are $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a_0 & +1.85576 & 3.26\times 10^{-8} & \{+1.85576,+1.85576\} \\ a_1 & +1.03359 & 8.54\times 10^{-6} & \{+1.03358,+1.03361\} \\ a_2 & -0.00601 & 7.72\times 10^{-5} & \{-0.00616,-0.00586\} \\ a_3 & +0.02915 & 2.40\times 10^{-4} & \{+0.02867,+0.02963\} \\ a_4 & -0.08568 & 2.44\times 10^{-4} & \{-0.08616,-0.08519\} \\ \end{array}$$ Over the considered range the error is less than $10^{-6}$.