TY - JOUR
AU - Nguyen, T. H.
AU - Vishnyakova, A.
PY - 2021/12/26
Y2 - 2023/06/09
TI - On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients
JF - Matematychni Studii
JA - Mat. Stud.
VL - 56
IS - 2
SE - Articles
DO - 10.30970/ms.56.2.149-161
UR - http://matstud.org.ua/ojs/index.php/matstud/article/view/253
SP - 149-161
AB - For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= \frac{a_{k-1}^2}{a_{k-2}a_k}, k \geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k \in \mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P\'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.
ER -