Hardy's generalization of \(e^z\) and related analogs of cosine and sine (Q2497123)

In 1904, Hardy introduced a family of entire functions of finite degree \[ E_{s,a}(z)=\sum_{n=0}^{\infty}(n+a)^s\frac{z^n}{n!}, \] hence \(E_{0,a}(z)=e^z\), and studied their asymptotic behavior and the asymptotic distribution of their zeros. The present author continues on the study of the functions \(E_{s,a}\), thus commemorating the centennial of Hardy's paper. He introduces natural generalizations of \(\sin z\) and \(\cos z\) as \[ S_{s,a}(z):=\frac{1}{2\imath}\left(E_{s,a}(\imath z)-E_{s,a}(-\imath z)\right) \] and \[ C_{s,a}(z):=\frac{1}{2}\left(E_{s,a}(\imath z)+E_{s,a}(-\imath z)\right), \] and skillfully studies the distribution of their zeros. We cite two typical results, here \[ G^+:=\{(s,a): \;s\geq 0,a>0\}\cup \{(s,a)\neq (-1,1):\;-1\leq s\leq 0,a\geq 1\}, \] \[ G^-:=\{(s,a)\neq (-1,1):\;s\leq -1, 0<a\leq 1\}. \] Theorem 2.3. (i) If \((s,a)\in G^+,\gamma \leq 1/2\), then the zeros of functions \[ C_{s,a}(z)\cos \gamma z+S_{s,a}(z)\sin \gamma z, \quad S_{s,a}(z)\cos \gamma z-C_{s,a}(z)\sin \gamma z \] are real, simple, and interlace. (ii) If \((s,a)\in G^-,\gamma \geq 1/2\), then also the zeros of these functions are real, simple, and interlace. Theorem 2.5 (conjectured by Moshe Newman). (i) For \(k<s\leq k+1,k=0,1,2,\ldots ,\) the set of all real zeros of \(E_{s,a}\) consists of \(k+1\) negative simple zeros. (ii) For \(s<0\) the function \(E_{s,a}\) does not have real zeros.