§12.11 Zeros

If $a\ge -\frac{1}{2}$, then $U(a,x)$ has no real zeros. If , then $U(a,x)$ has no positive real zeros. If , $n=1,2,\mathrm{\dots }$, then $U(a,x)$ has $n$ positive real zeros. Lastly, when $a=-n-\frac{1}{2}$, $n=1,2,\mathrm{\dots }$ (Hermite polynomial case) $U(a,x)$ has $n$ zeros and they lie in the interval $[-2\sqrt{-a},2\sqrt{-a}]$. For further information on these cases see Dean (1966).

If $a>-\frac{1}{2}$, then $V(a,x)$ has no positive real zeros, and if $a=\frac{3}{2}-2n$, $n\in \mathbb{Z}$, then $V(a,x)$ has a zero at $x=0$.

When $a>-\frac{1}{2}$, $U(a,z)$ has a string of complex zeros that approaches the ray $\mathrm{ph}z=\frac{3}{4}\pi$ as $z\to \mathrm{\infty }$, and a conjugate string. When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and

with

and

When $a=\frac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\mathrm{erfc}(z/\sqrt{2})$; compare (12.7.5). Numerical calculations in this case show that $z_{\frac{1}{2},s}$ corresponds to the $s$th zero on the string; compare §7.13(ii).

For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. Then

as $\mu$ ($=\sqrt{-2a}$) $\to \mathrm{\infty }$, $s$ fixed. Here $\alpha ={\mu }^{-\frac{4}{3}}{a}_s$, $a_s$ denoting the $s$th negative zero of the function $\mathrm{Ai}$ (see §9.9(i)). The first two coefficients are given by

where $t(\zeta )$ is the function inverse to $\zeta (t)$, defined by (12.10.39) (see also (12.10.41)), and

Similarly, for the zeros of $U^{\prime }(a,x)$ we have

where $\beta ={\mu }^{-\frac{4}{3}}{a}_s^{\prime }$, $a_s^{\prime }$ denoting the $s$th negative zero of the function ${\mathrm{Ai}}^{\prime }$ and

For the first zero of $U(a,x)$ we also have

where the numerical coefficients have been rounded off.

For further information, including associated functions, see Olver (1959).