Let , , denote the zeros of with
Then is strictly increasing in and strictly decreasing in ; furthermore, if , then is strictly increasing in .
Also, with defined as in (18.15.5)
except when .
Let be the th positive zero of the Bessel function (§10.21(i)). Then
Let . Then as , with () and () fixed,
uniformly for , where is an arbitrary constant such that .
See Dimitrov and Nikolov (2010).
For ultraspherical and Legendre polynomials, set and , respectively, in the results given in §18.16(ii).
The zeros of are denoted by , , with
Also, is again defined by (18.15.17).
For , and with as in §18.16(ii),
The constant in (18.16.10) is the best possible since the ratio of the two sides of this inequality tends to 1 as .
For the smallest and largest zeros we have
All zeros of lie in the open interval . In view of the reflection formula, given in Table 18.6.1, we may consider just the positive zeros , . Arrange them in decreasing order:
where is the th negative zero of (§9.9(i)), , and as with fixed
For an asymptotic expansion of as that applies uniformly for , see Olver (1959, §14(i)). In the notation of this reference , , and . For an error bound for the first approximation yielded by this expansion see Olver (1997b, p. 408).