# §14.16 Zeros

## §14.16(i) Notation

Throughout this section we assume that and are real, and when they are not integers we write

where , and , . For all cases concerning and we assume that without loss of generality (see (14.9.5) and (14.9.11)).

## §14.16(ii) Interval

The number of zeros of in the interval is if any of the following sets of conditions hold:

• (a)

.

• (b)

, , and .

• (c)

, , and is odd.

• (d)

.

The number of zeros of in the interval is if either of the following sets of conditions holds:

• (a)

, , and .

• (b)

, , and is even.

The zeros of in the interval interlace those of . has zeros in the interval , where can take one of the values −1, 0, 1, 2, subject to being even or odd according as and have opposite signs or the same sign. In the special case and , has zeros in the interval .

For uniform asymptotic approximations for the zeros of in the interval when with fixed, see Olver (1997b, p. 469).

## §14.16(iii) Interval

has exactly one zero in the interval if either of the following sets of conditions holds:

• (a)

, , , and and have opposite signs.

• (b)

, , and is odd.

For all other values of and (with ) has no zeros in the interval .

has no zeros in the interval when , and at most one zero in the interval when .