On a Class of Sine-Type Functions

We study an infinite product $F_\Lambda(z)$ with zeros $\lambda_n=n+l(|n|)$, $n\in\mathbb Z$, where $l(t)$ is a concave function and $l(t)=o(t)$. We obtain a test for $F_\Lambda(z)$ to belong to the class of sine-type functions. For the particular case in which $l(t)$ is a regularly varying function, we obtain sharp asymptotic estimates for $F_\Lambda(z)$.