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

M. Almohamed ; I. V. Tikhonov ; V. B. Sherstyukov

Vladikavkazskij matematičeskij žurnal, Tome 27 (2025) no. 1, pp. 5-20

Résumé

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.