scipy.special.

factorialk#

scipy.special.factorialk(n, k, exact=False, extend='zero')[source]#

Multifactorial of n of order k, n(!!…!).

This is the multifactorial of n skipping k values. For example,

factorialk(17, 4) = 17!!!! = 17 * 13 * 9 * 5 * 1

In particular, for any integer n, we have

factorialk(n, 1) = factorial(n)

factorialk(n, 2) = factorial2(n)

Parameters:

nint or float or complex (or array_like thereof): Input values for multifactorial. Non-integer values require extend='complex'. By default, the return value for n < 0 is 0.
nint or float or complex (or array_like thereof): Order of multifactorial. Non-integer values require extend='complex'.
exactbool, optional: If exact is set to True, calculate the answer exactly using integer arithmetic, otherwise use an approximation (faster, but yields floats instead of integers) Default is False.
extendstring, optional: One of 'zero' or 'complex'; this determines how values n<0 are handled - by default they are 0, but it is possible to opt into the complex extension of the multifactorial. This enables passing complex values, not only to n but also to k.

Warning

Using the 'complex' extension also changes the values of the multifactorial at integers n != 1 (mod k) by a factor depending on both k and n % k, see below or [1].

Returns:

nfint or float or complex or ndarray: Multifactorial (order k) of n, as integer, float or complex (depending on exact and extend). Array inputs are returned as arrays.

Notes

While less straight-forward than for the double-factorial, it’s possible to calculate a general approximation formula of n!(k) by studying n for a given remainder r < k (thus n = m * k + r, resp. r = n % k), which can be put together into something valid for all integer values n >= 0 & k > 0:

n!(k) = k ** ((n - r)/k) * gamma(n/k + 1) / gamma(r/k + 1) * max(r, 1)

This is the basis of the approximation when exact=False.

In principle, any fixed choice of r (ignoring its relation r = n%k to n) would provide a suitable analytic continuation from integer n to complex z (not only satisfying the functional equation but also being logarithmically convex, c.f. Bohr-Mollerup theorem) – in fact, the choice of r above only changes the function by a constant factor. The final constraint that determines the canonical continuation is f(1) = 1, which forces r = 1 (see also [1]).:

z!(k) = k ** ((z - 1)/k) * gamma(z/k + 1) / gamma(1/k + 1)

References

[1]

Complex extension to multifactorial https://en.wikipedia.org/wiki/Double_factorial#Alternative_extension_of_the_multifactorial

Examples

>>> from scipy.special import factorialk
>>> factorialk(5, k=1, exact=True)
120
>>> factorialk(5, k=3, exact=True)
10
>>> factorialk([5, 7, 9], k=3, exact=True)
array([ 10,  28, 162])
>>> factorialk([5, 7, 9], k=3, exact=False)
array([ 10.,  28., 162.])