On zeros of the combination of products of Bessel functions

In this paper, the function $f_\nu(t)=J_\nu(t)I_{-\nu}(t)+I_\nu(t)J_{-\nu}(t)$, $0<\nu<1$, $\mathrm{Re}\,t>0$, is investigated. Such functions were little studied in the literature. It is proved that more general functions $f_{\nu,\mu}^{(1),(2)}(t)=J_\nu(t)I_{-\mu}(t)\pm I_\mu(t)J_{-\nu}(t)$ have a countable set of real zeros and a countable set of pure imaginary zeros. The proof uses the well-known Sturm theorem for second-order differential equations. The statement is applied to specific examples. In the case $\nu=1/2$, the function $f_{1/2}(x)=J_{1/2}(x)I_{-1/2}(x)+I_{1/2}(x)J_{-1/2}(x)$ is reduced to an elementary function $f_{1/2}(x)=\frac2{\pi x}(\sin x\cdot\cosh x+\cos x\cdot\sinh x)$, and an asymptotic formula for its positive zeros $x=-\frac\pi4+\pi k+O(e^{-2\pi k})$ is found. Function $\hat{f}_{1/2}(x)=J_{1/2}(x)I_{-1/2}(x)-I_{1/2}(x)J_{-1/2}(x)$ has the following positive zeros: $x=\frac\pi4+\pi k+O(e^{-2\pi k})$.