The Pochhammer symbol 
denotes the rising factorial in the theory of
special functions. For a complex
number 
and a nonnegative integer 
, it is defined by
 |  |  |
(1)
|
 |  |  |
(2)
|
(Abramowitz and Stegun 1972, p. 256; Spanier and Oldham 1987; Koepf 1998, p. 5), where the product is 1 when there are no factors. The
gamma function quotient is interpreted by continuity
at nonpositive integer 
when 
is a nonnegative integer.
It also defines 
for other orders wherever the quotient is well defined. Other names are rising factorial
power (Graham et al. 1994, p. 48) and ascending factorial (Boros and
Moll 2004, p. 16). The Pochhammer symbol is implemented in the Wolfram
Language as Pochhammer[x,
n].
In combinatorics, the notation 
(Roman 1984, p. 5), 
(Comtet 1974, p. 6), or 
(Graham et al. 1994, p. 48) is used for
the rising factorial, while 
or 
denotes the falling
factorial (Graham et al. 1994, p. 48). Extreme caution is therefore
needed in interpreting the notations 
and 
.
The first few values of 
for nonnegative integers 
are
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
(OEIS A054654).
In closed form, 
can be written
 |
(8)
|
where 
is a Stirling number of the first kind.
The Pochhammer symbol satisfies
 |
(9)
|
the dimidiation formulas
 |  |  |
(10)
|
 |  |  |
(11)
|
and the duplication formula
 |
(12)
|
(Boros and Moll 2004, p. 17).
For nonnegative integers 
and 
with 
, a ratio of Pochhammer symbols is given in closed
form by
 |
(13)
|
(Boros and Moll 2004, p. 17).
For nonnegative integer 
, the derivative is given by
![d/(dx)(x)_n=(x)_n[psi_0(n+x)-psi_0(x)],](/images/equations/PochhammerSymbol/NumberedEquation5.svg) |
(14)
|
where 
is the digamma function and the right side is
interpreted by continuity at its removable
singularities.
Special values include
 |  |  |
(15)
|
 |  |  |
(16)
|
The Pochhammer symbol obeys the Euler transformation (Nørlund 1955, p. 294) as an identity of formal power series
 |
(17)
|
where 
is the forward difference and
 |
(18)
|
(Nørlund 1955).
For a positive integer 
and a real number 
, the sum of 
is given in closed form
by
 |
(19)
|
using the gamma function quotient to define 
.
Consider the finite product
 |  |  |
(20)
|
 |  |  |
(21)
|
where 
is a positive integer. Define
 |
(22)
|
using the principal branch of the natural logarithm. For 
outside the real interval ![[-1,0]](/images/equations/PochhammerSymbol/Inline69.svg)
, the Euler-Maclaurin
integration formulas give
 |
(23)
|
The critical curve is given by the implicit equation
 |
(24)
|
Inside this curve, 
converges to 0, whereas outside it, its absolute
value tends to infinity. The rightmost real point
is 
(OEIS A090462), and the leftmost is 
. The real convergence
set is ![(x_-,x_+]](/images/equations/PochhammerSymbol/Inline73.svg)
.
At 
,
the limit is
 |
(25)
|
whereas at 
the values alternate in sign and do not converge. The extremal imaginary coordinates
of the critical curve are 
(OEIS A090463),
occurring at 
.
On the critical curve,
 |
(26)
|
but 
generally does not converge because its complex argument
oscillates.
Plotting a suitably scaled version of 
shows beautiful and subtle structures such as those illustrated
above for 
(M. Trott, pers. comm., Dec. 1, 2003).
Another beautiful visualization plots 
, as illustrated above for 
(M. Trott, pers. comm., Dec. 2, 2003).
." Ch. 18 in