Falling and rising factorials

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial

(
                x
                
                  )
                  
                    n
                  
                
                =
                
                  x
                  
                    
                      n
                      _
                    
                  
                
              
              
                
                =
                
                  
                    
                      
                        x
                        (
                        x
                        −
                        1
                        )
                        (
                        x
                        −
                        2
                        )
                        ⋯
                        (
                        x
                        −
                        n
                        +
                        1
                        )
                      
                      ⏞
                    
                  
                  
                    n
                    
                       factors
                    
                  
                
              
            
            
              
              
                
                =
                
                  ∏
                  
                    k
                    =
                    1
                  
                  
                    n
                  
                
                (
                x
                −
                k
                +
                1
                )
                =
                
                  ∏
                  
                    k
                    =
                    0
                  
                  
                    n
                    −
                    1
                  
                
                (
                x
                −
                k
                )
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(x)_{n}=x^{\underline {n}}&=\overbrace {x(x-1)(x-2)\cdots (x-n+1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x-k+1)=\prod _{k=0}^{n-1}(x-k).\end{aligned}}}

The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as

x
                  
                    (
                    n
                    )
                  
                
                =
                
                  x
                  
                    
                      n
                      ¯
                    
                  
                
              
              
                
                =
                
                  
                    
                      
                        x
                        (
                        x
                        +
                        1
                        )
                        (
                        x
                        +
                        2
                        )
                        ⋯
                        (
                        x
                        +
                        n
                        −
                        1
                        )
                      
                      ⏞
                    
                  
                  
                    n
                    
                       factors
                    
                  
                
              
            
            
              
              
                
                =
                
                  ∏
                  
                    k
                    =
                    1
                  
                  
                    n
                  
                
                (
                x
                +
                k
                −
                1
                )
                =
                
                  ∏
                  
                    k
                    =
                    0
                  
                  
                    n
                    −
                    1
                  
                
                (
                x
                +
                k
                )
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}x^{(n)}=x^{\overline {n}}&=\overbrace {x(x+1)(x+2)\cdots (x+n-1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x+k-1)=\prod _{k=0}^{n-1}(x+k).\end{aligned}}}

The value of each is taken to be 1 (an <a href="/facts/Empty_product/mj7nhGeD">empty product</a>) when 
 
 
 
 n
 =
 0
 
 
 {\displaystyle n=0}
 
. These symbols are collectively called factorial powers.
The Pochhammer symbol, introduced by <a href="/facts/Leo_August_Pochhammer/R74KVmoe">Leo August Pochhammer</a>, is the notation 
 
 
 
 (
 x
 
 )
 
 n
 
 
 
 
 {\displaystyle (x)_{n}}
 
, where n is a <a href="/facts/Non-negative_integer/u65og8JE">non-negative integer</a>. It may represent either the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used 
 
 
 
 (
 x
 
 )
 
 n
 
 
 
 
 {\displaystyle (x)_{n}}
 
 with yet another meaning, namely to denote the <a href="/facts/Binomial_coefficient/Tsx7eY5h">binomial coefficient</a> 
 
 
 
 
 
 
 
 (
 
 
 x
 n
 
 
 )
 
 
 
 
 
 
 {\displaystyle {\tbinom {x}{n}}}
 
.
In this article, the symbol 
 
 
 
 (
 x
 
 )
 
 n
 
 
 
 
 {\displaystyle (x)_{n}}
 
 is used to represent the falling factorial, and the symbol 
 
 
 
 
 x
 
 (
 n
 )
 
 
 
 
 {\displaystyle x^{(n)}}
 
 is used for the rising factorial. These conventions are used in <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>,
although <a href="/facts/Donald_Knuth/GozNzDM6">Knuth</a>'s underline and overline notations 
 
 
 
 
 x
 
 
 n
 _
 
 
 
 
 
 {\displaystyle x^{\underline {n}}}
 
 and 
 
 
 
 
 x
 
 
 n
 ¯
 
 
 
 
 
 {\displaystyle x^{\overline {n}}}
 
 are increasingly popular.
In the theory of <a href="/facts/Special_functions/JETSeucA">special functions</a> (in particular the <a href="/facts/Hypergeometric_function/yTEC6NBJ">hypergeometric function</a>) and in the standard reference work <a href="/facts/Abramowitz_and_Stegun/CNGSV3Ed">Abramowitz and Stegun</a>, the Pochhammer symbol 
 
 
 
 (
 x
 
 )
 
 n
 
 
 
 
 {\displaystyle (x)_{n}}
 
 is used to represent the rising factorial.
When 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is a positive integer, 
 
 
 
 (
 x
 
 )
 
 n
 
 
 
 
 {\displaystyle (x)_{n}}
 
 gives the number of <a href="/facts/K-permutation/JSw9ukmv">n-permutations</a> (sequences of distinct elements) from an x-element set, or equivalently the number of <a href="/facts/Injective_function/5ES6tqf5">injective functions</a> from a set of size 
 
 
 
 n
 
 
 {\displaystyle n}
 
 to a set of size 
 
 
 
 x
 
 
 {\displaystyle x}
 
. The rising factorial 
 
 
 
 
 x
 
 (
 n
 )
 
 
 
 
 {\displaystyle x^{(n)}}
 
 gives the number of <a href="/facts/Partition_of_a_set/VeP9g8CA">partitions</a> of an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-element set into 
 
 
 
 x
 
 
 {\displaystyle x}
 
 ordered sequences (possibly empty).

Falling and rising factorials open-in-new

Falling and rising factorials