§28.9 Zeros

For real $q$ each of the functions $\mathop{\mathrm{ce}_{2n}\/}\nolimits\!\left(z,q\right)$, $\mathop{\mathrm{se}_{2n+1}\/}\nolimits\!\left(z,q\right)$, $\mathop{\mathrm{ce}_{2n+1}\/}\nolimits\!\left(z,q\right)$, and $\mathop{\mathrm{se}_{2n+2}\/}\nolimits\!\left(z,q\right)$ has exactly $n$ zeros in $0. They are continuous in $q$. For $q\to\infty$ the zeros of $\mathop{\mathrm{ce}_{2n}\/}\nolimits\!\left(z,q\right)$ and $\mathop{\mathrm{se}_{2n+1}\/}\nolimits\!\left(z,q\right)$ approach asymptotically the zeros of $\mathop{\mathit{He}_{2n}\/}\nolimits\!\left(q^{1/4}(\pi-2z)\right)$, and the zeros of $\mathop{\mathrm{ce}_{2n+1}\/}\nolimits\!\left(z,q\right)$ and $\mathop{\mathrm{se}_{2n+2}\/}\nolimits\!\left(z,q\right)$ approach asymptotically the zeros of $\mathop{\mathit{He}_{2n+1}\/}\nolimits\!\left(q^{1/4}(\pi-2z)\right)$. Here $\mathop{\mathit{He}_{n}\/}\nolimits\!\left(z\right)$ denotes the Hermite polynomial of degree $n$18.3). Furthermore, for $q>0$ $\mathop{\mathrm{ce}_{m}\/}\nolimits\!\left(z,q\right)$ and $\mathop{\mathrm{se}_{m}\/}\nolimits\!\left(z,q\right)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of $\mathop{J_{m}\/}\nolimits\!\left(2\sqrt{q}\mathop{\cos\/}\nolimits z\right)$10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\Im{z}\right|\to\infty$. There are no zeros within the strip $\left|\Re{z}\right|<\tfrac{1}{2}\pi$ other than those on the real and imaginary axes.

For further details see McLachlan (1947, pp. 234–239) and Meixner and Schäfke (1954, §§2.331, 2.8, 2.81, and 2.85).