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Pochhammer






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Gamma, Beta, Erf > Pochhammer[a,n] > Introduction to the factorials and binomials





The best-known properties and formulas for factorials and binomials

Real values for real arguments

For real values of arguments, the values of the factorials and binomials , , , , and are real (or infinity).

Simple values at zero

The factorials and binomials , , , , and have simple values for zero arguments:

Values at fixed points

Students usually learn the following basic table of values of the factorials and in special integer points:

Specific values for specialized variables

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:

Analyticity

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 .

Poles and essential singularities

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 .

Periodicity

The factorials and binomials , , , , and do not have periodicity.

Parity and symmetry

The factorials and binomials , , , , and have mirror symmetry:

The multinomial has permutation symmetry:

Series representations

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:

Asymptotic series expansions

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:

Integral representations

The factorial and binomial can also be represented through the following integrals:

Transformations

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:

Identities

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:

Representations of derivatives

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 :





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