On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients
Анотація
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 zero
with 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.
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