[bug] torch.polygamma inconsistent with scipy.special.polygamma for n >= 1

[kshitij12345]

n = 1

>>> t = torch.tensor([-1.])
>>> torch.polygamma(1, t)
tensor([1.2914e+15])
>>> scipy.special.polygamma(1, t.numpy())
array([inf], dtype=float32)

>>> t = torch.tensor([-501.])
>>> torch.polygamma(1, t)
tensor([2.0831e+09])
>>> scipy.special.polygamma(1, t.numpy())
array([inf], dtype=float32)

>>> t = torch.tensor([-float('inf')])
>>> torch.polygamma(1, t)
tensor([nan])
>>> scipy.special.polygamma(1, t.numpy())
array([inf], dtype=float32)
>>> 

n > 1

>>> t = torch.tensor([float('inf')])
>>> torch.polygamma(2, t)
tensor([nan])
>>> scipy.special.polygamma(2, t.numpy())
array([-0.], dtype=float32)
>>> torch.polygamma(3, t)
tensor([nan])
>>> scipy.special.polygamma(3, t.numpy())
array([0.], dtype=float32)
>>> torch.polygamma(4, t)
tensor([nan])
>>> scipy.special.polygamma(4, t.numpy())
array([-0.], dtype=float32)

cc @mruberry

[mruberry]

For the case where n=1, are we computing the correct values?

[kshitij12345]

I am planning to verify that and fix this issue in separate PR once #51966 is in

[mruberry]

Sounds good. Next question: for n > 2 are we only divergent for inf or other extremals, or also for real values?

[kshitij12345]

Only the extremals. (However on ROCM we are divergent on real values as well)

[kshitij12345]

Reopening as #61641 was reverted.

[albanD]

Reland happened

[igm503]

The inconsistency between torch and scipy (+ the actual value of polygamma_1 at negative integers) seems to remain as of the latest release of torch and the latest nightly.

@albanD Do you know why was this closed? Was there a decision to leave the current torch behavior in place, or is it just that no one remembered to reopen when the reland was reverted? (I'm assuming it was reverted, since I don't see any of the changes in the current codebase, and the polygamma behavior remains, at least for negative ints). Here's the commit I'm talking about: #62322