I am interested in the monotonicity properties of the Cylindrical Bessel Function (CBF) of the 1st kind with integer order $n$ and real argument $\beta$ or $J_n\{\beta\}$. What I'm hoping exists somewhere is a relatively simple algebraic expression or sequence as a function of $n$ and $\beta$ that I can evaluate rather than trying to use tables of Bessel functions or an exhaustive computation in say MATLAB.
Through my literature search I'm familiar with the bounds derived by Landau [1] which gives an inequality on the bounds and monotonicity of the CBF and also with the results by Watson [2, pg. 303] which expresses the ratio of $\frac{J_{\nu+1}\{\beta\}}{J_{\nu}\{\beta\}}$ (where the order $\nu$ is a real number and not necessarily an integer) in terms of the Lommel Polynomials. I'm wondering if there are other results and methods in addition to these?
Another comment that may help you answer my question: It seems to me knowing something about the monotonicity of the CBF also provides insight in the rate of convergence of the identities $\sum_{n=-\infty}^{\infty}J_n\{\beta\} = 1$ and $\sum_{n=-\infty}^{\infty}J_n^2\{\beta\} = 1$. Further insight into the rate of convergence of these identities is one of the main reasons for my interest in this problem.
Any insights or references regarding this problem are greatly appreciated. Thanks in advance!
References:
[1] L. J. Landau, "Bessel Functions: Monotonicity and Bounds", J. London Mathematical Society, vol. 61, issue. 1, pp. 197-215, 2000.
[2] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, 1944.