Falling and rising factorial

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In mathematics, the falling or rising factorial denotes a function similar to exponentiation , but in which the factors fall or rise gradually, i.e. i.e., be reduced or increased by one.


For natural numbers and with the -th decreasing or increasing factorial is noted as or (in some older textbooks also or ) and is defined as follows:

Combinatorial interpretation

In the urn model , the falling factorial can be interpreted as the number of possibilities to remove balls from an urn with different balls , without replacing them, taking into account the sequence. There are candidates for the first ball , for the second ... and finally for the last ball . There are therefore options for the overall selection .

In general, the number of - permutations of a - set or, alternatively, the number of injective mappings of a - set into a - set.


The definition is analogous for a complex number and a natural number :

One can then understand and as complex polynomials in .

For the increasing factorial corresponds to the Pochhammer symbol .


Calculation rules

The following calculation rules apply:


Relationships with other known numbers

With the help of the falling factorials, the binomial coefficients can be defined in general:

The following equations also apply, where and denote the (unsigned) Stirling numbers of the first and second kind:

Occurrence in analysis