|
For real values of arguments, the values of the factorials and binomials  ,  ,  ,  , and  are real (or infinity).
The factorials and binomials  ,  ,  ,  , and  have simple values for zero arguments:
Students usually learn the following basic table of values of the factorials  and  in special integer points:
If variable  is a rational or integer number, the factorials  and  can be represented by the following general formulas:
For some particular values of the variables, the Pochhammer symbol  has the following meanings:
Some well‐known formulas for binomial and multinomial functions are:
The factorials and binomials  ,  ,  ,  , and  are defined for all complex values of their variables. The factorials, binomials, and multinomials are analytical functions of their variables and do not have branch cuts and branch points. The functions  and  do not have zeros:  ;  . Therefore, the functions  and  are entire functions with an essential singular point at  .
The factorials and binomials  ,  ,  ,  , and  have an essential singularity for infinite values of any argument. This singular point is also the point of convergence of the poles (except  for  ).
The function  has an infinite set of singular points:  are the simple poles with residues  .
The function  has an infinite set of singular points:  are the simple poles with residues  .
For fixed  , the function  has an infinite set of singular points:  are the simple poles with residues  .
For fixed  , the function  has an infinite set of singular points:  are the simple poles with residues  .
For fixed  , the function  has an infinite set of singular points:  are the simple poles with residues  .
By variable  (with the other variables fixed) the function  has an infinite set of singular points:  are the simple poles with residues
 .
The factorials and binomials  ,  ,  ,  , and  do not have periodicity.
The factorials and binomials  ,  ,  ,  , and  have mirror symmetry:
The multinomial  has permutation symmetry:
The factorials  ,  , and  have the following series expansions in the regular points:
The series expansions of  and  near singular points are given by the following formulas:
The asymptotic behavior of the factorials and binomials  ,  ,  ,  ,  can be described by the following formulas (only the main terms of asymptotic expansion are given). The first is the famous Stirling's formula:
The factorial  and binomial  can also be represented through the following integrals:
The following formulas describe some of the main types of transformations between and among factorials and binomials:
Some of these transformations can be called addition formulas, for example:
Multiple argument transformations are, for example:
The following transformations are for products of the functions:
The factorials  and  can be defined as the solutions of the following corresponding functional equations:
The factorial  is the unique nonzero solution of the functional equation  that is logarithmically convex for all real  ; that is, for which  is a convex function for  .
The factorials and binomials  ,  ,  ,  , and  satisfy the following recurrence identities:
The previous formulas can be generalized to the following recurrence identities with a jump of length n:
The Pochhammer symbol  and binomial  satisfy the following functional identities:
The derivatives of the functions  ,  ,  ,  ,  have rather simple representations that include the corresponding functions as factors:
The symbolic derivatives of the  order form factorials and binomials  ,  ,  ,  , and  have much more complicated representations, which can include recursive function calls, regularized generalized hypergeometric functions  , or Stirling numbers  :
|
