On multiple zeros of one entire function which is of interest for the theory of inverse problems

We consider complex zeros of one entire function from the theory of linear inverse problems for second-order differential equations. This function of order $ \rho=1/2 $ is elementary, transcendental, and depends in a simple way on a complex parameter $ p\in\mathbb{C}\setminus\{0\}$. It is required to find out whether there are values of $ p $ for which the function has multiple zeros. The question posed has been fully answered. It is shown that there exists a countable set of values $ p=p_n$, for each of which the entire function has not only an infinite number of simple zeros, but also one zero of multiplicity two. A description is given of both the set of such values $p_n$ and the corresponding multiple zeros. Our main result is expressed in terms of roots of the transcendental equation $\mathrm{sh}\, z=z$, the analysis of which is the subject of the final section of the paper. Here we announce new non-asymptotic estimates, applicable to all roots of the equation in the domain $ z\ne 0 $ and giving very precise localization for them. Numerical calculations confirm our analytical conclusions. There are useful connections with the theory of Mittag-Leffler functions and some spectral problems from mathematical physics.