The discrete Bessel kernel

Theorem 2 in a paper by Borodin, Okounkov, and Olshanski states that the discrete Bessel kernel $J(x,y,\theta)$ is given by

\begin{equation*} \sqrt{\theta} \frac{J_x J_{y+1} - J_{x+1} J_y}{x-y} \end{equation*}

where $J_x = J_x(2\sqrt{\theta})$ is the Bessel function of order $x$ and argument $2\sqrt{\theta}$.

Does anybody have any ideas on how to determine the asymptotics of this kernel in the case $x = \alpha \sqrt{n}, y = \beta \sqrt{n}$ where $\alpha,\beta$ are distinct real numbers that differ from $\pm 2$?

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You could start with the asymptotics for $J_x$ and apply trig identities to your kernel.