# Fix hurwitz_zeta(0, x) not expanding

Reported by: Owned by: pluschny minor sage-9.4 symbolics hurwitz_zeta slelievre Frédéric Chapoton Peter Luschny N/A 785b4e2 785b4e27162892519d6bb1d5208551893e28a52e

Having defined:

```def B(n, x):
return -n*hurwitz_zeta(1 - n, x) if n != 0 else 1
```

we expect `expand(B(n, x))` to match `bernoulli_polynomial(x, n)`.

Using this small utility function to compare them:

```def table(nmax):
nab = [('n', 'expand(B(n, x))', 'bernoulli_polynomial(x, n)')]
nab.extend([(n, expand(B(n, x)), bernoulli_polynomial(x, n))
for n in range(nmax + 1)])
k = len(str(nmax))
m = max(len(str(x)) for t in nab for x in t)
hline = '-' * (k + m + 7)
print(hline)
for n, a, b in nab:
print(f' %{k}s     %{m}s' % (n, a))
print(f' %{k}s     %{m}s' % ('', b))
print(hline)
```

we notice they match except for n = 1:

```sage: table(4)
----------------------------------
n                expand(B(n, x))
bernoulli_polynomial(x, n)
----------------------------------
0                              1
1
----------------------------------
1            -hurwitz_zeta(0, x)
x - 1/2
----------------------------------
2                  x^2 - x + 1/6
x^2 - x + 1/6
----------------------------------
3          x^3 - 3/2*x^2 + 1/2*x
x^3 - 3/2*x^2 + 1/2*x
----------------------------------
4       x^4 - 2*x^3 + x^2 - 1/30
x^4 - 2*x^3 + x^2 - 1/30
----------------------------------
```

This ticket fixes that case.

### comment:1 Changed 14 months ago by chapoton

• Authors set to Frédéric Chapoton
• Branch set to u/chapoton/32016
• Commit set to 785b4e27162892519d6bb1d5208551893e28a52e
• Status changed from new to needs_review

New commits:

 ​785b4e2 `Hurwitz zeta at n=0`

### comment:2 Changed 14 months ago by pluschny

• Status changed from needs_review to positive_review

Looks good to me. Merci!

### comment:4 Changed 14 months ago by pluschny

• Reviewers set to Peter Luschny

I don't know if I am qualified as a reviewer in the sense of the SageMath organization.

### comment:5 Changed 14 months ago by slelievre

• Description modified (diff)
• Summary changed from hurwitz_zeta(0, x) does not expand to Fix hurwitz_zeta(0, x) not expanding
• Type changed from PLEASE CHANGE to enhancement

I don't know if I am qualified as a reviewer in the sense of the SageMath organization.

You are!

### comment:6 Changed 14 months ago by slelievre

• Description modified (diff)

### comment:7 Changed 13 months ago by vbraun

• Branch changed from u/chapoton/32016 to 785b4e27162892519d6bb1d5208551893e28a52e
• Resolution set to fixed
• Status changed from positive_review to closed
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