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FactorialPower

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

FactorialPower[n, k]

The falling factorial n (n - 1) (n - 2) ... (n - k + 1). For non-negative integer k, expands to a product of k linear factors. Equivalent to n! / (n - k)! when both n and k are non-negative integers. For a non-integer numeric k (under N), evaluates as Gamma[n+1]/Gamma[n-k+1].

Examples (7)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (4)

In[1]:= FactorialPower[5, 3]
Out[1]= 60

In[2]:= FactorialPower[n, 3]
Out[2]= n (-2 + n) (-1 + n)

In[3]:= FactorialPower[n, 0]
Out[3]= 1

In[4]:= D[x^n, {x, k}]
Out[4]= FactorialPower[n, k] x^(-k + n)

Applications (3)

In[5]:= FactorialPower[10, 3]
Out[5]= 720

In[6]:= FactorialPower[x, 4]
Out[6]= x (-3 + x) (-2 + x) (-1 + x)

In[7]:= FactorialPower[10, 4] == 10!/6!
Out[7]= True

Implementation notes

builtin_factorialpower computes the falling factorial n^(k) = n(n-1)...(n-k+1). It requires k to be a concrete non-negative integer (k == 0 gives 1; negative or symbolic k returns NULL). For integer/bignum n it accumulates the exact product in GMP (mpz_mul of n-i), normalising via expr_bigint_normalize. For symbolic n with small k (<= 32) it expands the literal product Times[n, n-1, ..., n-k+1] through eval_and_free so Expand/D can act on it. The 1-argument step-form is not handled here.

  • Protected, Listable, NumericFunction.
  • For non-negative integer $n$ and non-negative integer $k$: exact GMP product.
  • For symbolic $n$ with concrete $k \le 32$: expands to an explicit product Times[n, n - 1, ..., n - k + 1] so Expand and D can act on it.
  • Equivalent to $n! / (n - k)!$ when both arguments are non-negative integers.
  • For a non-integer numeric $k$ (under N), evaluates the generalised form $\Gamma(n+1)/\Gamma(n-k+1)$ (real, complex, arbitrary precision); exact non-integer $k$ stays symbolic.
  • Used as the symbolic-order derivative of a power: $D[x^n, {x, k}] = \mathrm{FactorialPower}[n, k]\, x^{n-k}$.

Attributes: Listable, NumericFunction, Protected.

References

See also: Expand, D, N

Notes & additional examples

Notes

FactorialPower[n, k] is the falling factorial n (n-1) (n-2) ... (n-k+1), the product of k descending linear factors. For symbolic n it expands to that product, e.g. FactorialPower[x, 4] = x (x-1) (x-2) (x-3), the basis polynomial that makes the forward difference operator behave like differentiation (Δ x^(k) = k x^(k-1)). For non-negative integers it equals the ratio of factorials n! / (n-k)!, as the identity check confirms.