The zeros of entire functions

counteregms

Graduate student Rintaro “Yoshi” Yoshida will defend the degree of Ph.D. on Thursday May 2, 2:00pm, in Keller 301.

We invite the reader to consider the entire function whose restriction to $\mathbb R$ is pictured above:
$$\varphi_{_{1/5}}(x) = \sum_{k=0}^\infty \frac{x^k}{(k!)^{(6/5)}}$$
Do you think this function has any non-real zeros? Does it belong to the Laguerre-Pólya class? See the draft dissertation for answers.