Zusammenfassung für den Vortrag am 04.05.2022 (15:00 Uhr)
Seminar on Nonlinear AlgebraThu Hien Nguyen (V.N. Karazin Kharkiv National University)
On necessary and sufficient conditions for entire functions to belong to the Laguerre-Pólya class in terms of their Taylor coefficients
We discuss new necessary and sufficient conditions for the entire functions with positive Taylor coefficients to belong to the Laguerre–Pólya class. It is an important class of entire functions which are locally the limit of a sequence of real polynomials having only real zeros. For an entire function \(f(z) = \sum_{k=0}^{\infty} a_k z^k,\) we define the second quotients of Taylor coefficients as \(q_n(f) := \frac{a_{n-1}^2}{a_{n-2} a_{n}}, n\geq 2\) and find conditions on \(q_n(f)\) for \(f\) to belong to the Laguerre–Pólya class. Besides, we show the relation of the conditions to the partial theta function \(g_a =\sum _{k=0}^{\infty} \frac {z^k}{a^{k^2}}\), when \(a>1.\)