# §24.12 Zeros

## §24.12(i) Bernoulli Polynomials: Real Zeros

In the interval the only zeros of , , are , and the only zeros of , , are .

For the interval denote the zeros of by , , with

24.12.1

Then the zeros in the interval are .

When is even

24.12.2

and as with fixed,

24.12.4

When is odd , , and as with fixed,

24.12.5

Let be the total number of real zeros of . Then when , and

## §24.12(ii) Euler Polynomials: Real Zeros

For the interval denote the zeros of by , , with

24.12.7

Then the zeros in the interval are .

When is even , and as with fixed,

24.12.8

When is odd ,

24.12.9,
24.12.10,

and as with fixed,

24.12.11

## §24.12(iii) Complex Zeros

For complex zeros of Bernoulli and Euler polynomials, see Delange (1987) and Dilcher (1988). A related topic is the irreducibility of Bernoulli and Euler polynomials. For details and references, see Dilcher (1987b), Kimura (1988), or Adelberg (1992).

## §24.12(iv) Multiple Zeros

, , has no multiple zeros. The only polynomial with multiple zeros is .