Laguerre–Pólya class

Remarks

A function $f$ that is not identically zero belongs to the class if and only if $$f(z) = c\, z^m e^{-a z^2 + b z} \prod_{n} \left(1 - \frac{z}{x_n}\right) e^{z/x_n}.$$ Here $c, b \in \R$ with $c \ne 0$, $a \ge 0$, and $m \ge 0$ is an integer. The $x_n$ are nonzero real numbers with $\sum_n x_n^{-2} < \infty$, and the product is a Weierstrass product. Example: $e^{-z^2}$ is the limit of the polynomials $(1 - z^2/k)^k$, which have real coefficients and only the real roots $\pm\sqrt{k}$. Example: $\sin z = z \prod_{n \ge 1} (1 - z^2/(n^2\pi^2))$ is in the class. Non-example: $1 + z^2$ has real coefficients but the roots $\pm i$. It is not a limit of real-rooted real polynomials, because by Hurwitz's theorem such limits, when not identically zero, have only real zeros. Non-example: $e^{z^2}$ has no zeros, but its product form would need $a = -1 < 0$. The class is closed under differentiation.