plsquares SciMax Toolbox pochhammer_max_index

SciMax Toolbox >> pochhammer

pochhammer

Maxima Function

Calling Sequence

pochhammer (n, x)

Description

The Pochhammer symbol. For nonnegative integers n with n <= pochhammer_max_index, the expression pochhammer (x, n) evaluates to the product x (x + 1) (x + 2) ... (x + n - 1) when n > 0 and to 1 when n = 0. For negative n, pochhammer (x, n) is defined as (-1)^n / pochhammer (1 - x, -n). Thus

(%i1) pochhammer (x, 3);
(%o1)                   x (x + 1) (x + 2)
(%i2) pochhammer (x, -3);
                                 1
(%o2)               - -----------------------
                      (1 - x) (2 - x) (3 - x)

To convert a Pochhammer symbol into a quotient of gamma functions, (see Abramowitz and Stegun, equation 6.1.22) use makegamma; for example

(%i1) makegamma (pochhammer (x, n));
                          gamma(x + n)
(%o1)                     ------------
                            gamma(x)

When n exceeds pochhammer_max_index or when n is symbolic, pochhammer returns a noun form.

(%i1) pochhammer (x, n);
(%o1)                         (x)
                                 n
plsquares SciMax Toolbox pochhammer_max_index