If then has no real zeros. If , then has no positive real zeros. If , , then has positive real zeros. Lastly, when , (Hermite polynomial case) has zeros and they lie in the interval . For further information on these cases see Dean (1966).
If then has no positive real zeros, and if , , then has a zero at .
When , has a string of complex zeros that approaches the ray as , and a conjugate string. When the zeros are asymptotically given by and , where is a large positive integer and
For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the th real zeros of and , counted in descending order away from the point , be denoted by and , respectively. Then
as () , fixed. Here , denoting the th negative zero of the function (see §9.9(i)). The first two coefficients are given by
Similarly, for the zeros of we have
where , denoting the th negative zero of the function and
For the first zero of we also have
where the numerical coefficients have been rounded off.
For further information, including associated functions, see Olver (1959).