/** This program calculates the "rising factorial" and the "falling factorial."
    and the "double factorial."

    Sometimes one of these is called the "Pochhammer function" but that naming
    convention is almost certainly going to be ambiguous.  (See Knuth, below)

    Beware the (terrible) notation as (v)_k or v^(n) which may represent either
    the rising or falling factorial!  (Just because a quantity is in parentheses
    you're supposed to infer that it means some totally different thing than
    that number in parentheses?!  Get out of here with that, you, and your
    binomial theorem notation and your Legendre symbol notation and your Jacobi
    symbol notation and every other mathematician who thought "hey I'm just
    gonna make parentheses mean something TOTALLY NEW and AMBIGUOUS and name it
    after myself.")

    Pochhammer himself used that notation to mean something else!  So stop it!

    See
    Knuth, Donald E.  "Two Notes on Notation",
    https://arxiv.org/abs/math/9205211

    beginning around equation (2.11) for many more details and better (but still
    ambiguous and awful) notation.
*/


/** Calculate the rising factorial.

    The rising factorial calculates:

     v (v+1) (v+2) ... (v + k - 1)

    This function is closely related to the gamma function, which is closely
    related to the factorial.

    This implementation includes an extension to negative numbers, where:

    risingFactorial[v, -k] := (-1)^-k / risingFactorial[1-v, -k]

    Since this function only utilizes basic arithmetic, it automatically
    works to arbitrary precision and works for complex numbers.

    This function also works with symbolic values of v:

    risingFactorial[x,3]

  produces:
    x (1 + x) (2 + x)    

    See:
    https://en.wikipedia.org/wiki/Falling_and_rising_factorials

    Gonzalez, Ivan & Jiu, Lin & Moll, Victor. (2015). Pochhammer Symbol with
    Negative Indices. A New Rule for the Method of Brackets. Open
    Mathematics. 14. 10.1515/math-2016-0063.

   Which can be found at:

https://www.researchgate.net/publication/280695601_Pochhammer_Symbol_with_Negative_Indices_A_New_Rule_for_the_Method_of_Brackets/fulltext/55c4077508aeb97567402184/280695601_Pochhammer_Symbol_with_Negative_Indices_A_New_Rule_for_the_Method_of_Brackets.pdf?origin=publication_detail

   or in arxiv:
   https://arxiv.org/abs/1508.00056
*/
risingFactorial[v, k] :=
{
   if ! isInteger[k]
   {
      println["risingFactorial not defined for non-integer value k = $k"]
      return undef
   }

   // Extension to negative numbers
   if k < 0
      return (-1)^-k / risingFactorial[1-v, -k]
   
   product = 1
   for s = 0 to k-1
      product = product * (v+s)
   
   return product
}


/** An alternate name for the rising factorial.  As noted above, the
    name "Pochhammer" can indicate a rising or falling factorial depending
    on the conventions in your field and this usage should be deprecated.
*/
Pochhammer[v, k] := risingFactorial[v, k]


/** Calculate the falling factorial.

   This calculates
   x (x-1) (x-2) ... (x-n+1)    
    
   See
   https://en.wikipedia.org/wiki/Falling_and_rising_factorials
*/
fallingFactorial[x, n] :=
{
   product = 1
   for k = 0 to n-1
      product = product * (x-k)

   return product
}


/** This is a generalization of the binomial theorem to rational r using
    Newton's generalization.  r can even be symbolic.

    This utilizes the fact that the binomial theorem for (r, k)
    can be generalized for r and integer k by:

    fallingFactorial[r, k] / k!

    See:
    https://en.wikipedia.org/wiki/Binomial_theorem#Newton's_generalized_binomial_theorem
*/
generalizedBinomial[r, k] := fallingFactorial[r, k] / k!


/** The so-called "double factorial" which is sometimes represented as n!!
    which is terrible ambiguous wrong notation.

    See:
    https://en.wikipedia.org/wiki/Double_factorial
*/
doubleFactorial[n] :=
{
   product = 1
   for k = 0 to ceil[n/2] - 1
      product = product * (n - 2 k)

   return product
}