# §18.16 Zeros

See §18.2(vi).

## §18.16(ii) Jacobi

Let , , denote the zeros of with

18.16.1

Then is strictly increasing in and strictly decreasing in ; furthermore, if , then is strictly increasing in .

### ¶ Inequalities

Let be the th positive zero of the Bessel function 10.21(i)). Then

### ¶ Asymptotic Behavior

Let . Then as , with () and () fixed,

uniformly for , where is an arbitrary constant such that .

### ¶ Other Bounds

See Dimitrov and Nikolov (2010).

## §18.16(iii) Ultraspherical and Legendre

For ultraspherical and Legendre polynomials, set and , respectively, in the results given in §18.16(ii).

## §18.16(iv) Laguerre

The zeros of are denoted by , , with

18.16.9

Also, is again defined by (18.15.17).

### ¶ Inequalities

For , and with as in §18.16(ii),

18.16.11

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

18.16.12
18.16.13

### ¶ Asymptotic Behavior

As , with and fixed,

where is the th negative zero of 9.9(i)). For three additional terms in this expansion see Gatteschi (2002). Also,

when .

## §18.16(v) Hermite

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:

Then

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).

Lastly, in view of (18.7.19) and (18.7.20), results for the zeros of lead immediately to results for the zeros of .