Add workaround for scipy.special.factorial2 changes

[FarnazH]

This PR adds a factorial2 function that wraps scipy.special.factorial2 to return 1.0 when the input is not positive. This is a temporary addition, while we wait for Scipy's update. To learn more, see scipy/scipy#18409. This is a better solution than limiting Scipy version that works with IOData, as one might need to use scipy 1.11.x in their environment for other purposes. A similar change was made in other QC-Devs packages; see theochem/gbasis#138.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Comparison is base (ca6113b) 95.00% compared to head (feecb0b) 94.98%.

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Additional details and impacted files

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- Coverage   95.00%   94.98%   -0.02%     
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  Files          74       74              
  Lines        8844     8819      -25     
  Branches     1216     1216              
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- Hits         8402     8377      -25     
  Misses        193      193              
  Partials      249      249              

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[PaulWAyers]

Shouldn't we return 1 for factorial2(-1) and 0 for factorial2(x) where x = -2,-3,....? I think that is the "old" scipy behavior.

Should we check that n is an integer?

Addiiton: Looking at wikipedia and Wolfram Mathworld, double-factorials for negative numbers are a mess, though everyone agrees they are undefined for negative even integers. It would be good to return NotImplementedError or something like that except for negative odd integers, $n$, which can be evaluated as:

$$ n!! = \frac{(-1)^{\frac{|n|-1}{2}}|n|}{|n|!!} = \frac{(-1)^{\frac{|n|-1}{2}}}{|n-2|!!} $$

The advantage of the first formula is that it also applies for $n=-1$, while the second formula, although simpler, is only useful starting at $n=-3$. I don't know that $n=-3,-5,\ldots$ ever appears, but we should probably return NotImplementedError or otherwise a sensible value.

[FarnazH]

@PaulWAyers, thanks for your feedback.

• We don't need to check that n is an integer, because this is just a wrapper function, and scipy.special.factorial2 does that check and prints a proper error message: ValueError: factorial2 does not support non-integral scalar arguments.
• Following your suggestion, I updated the function to return 0.0 for n <= -2; see b3b7377. As scipy does not treat odd and even negative integers differently, I left it intact.