Falling Factorial Website Links For
Symbol 		 

Information About

Falling Factorial
  CATEGORIES ABOUT POCHHAMMER SYMBOL
   
gamma and related functions
   
factorial and binomial topics
   
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...



:(x)_n\,

is used in the theory of Special Functions to represent the "rising factorial" or "upper factorial"

:(x)_n=x(x+1)(x+2)\cdots(x+n-1)= rac{(x+n-1)!}{(x-1)!}

and, confusingly, is used in Combinatorics to represent the "falling factorial" or "lower factorial"

:(x)_n=x(x-1)(x-2)\cdots(x-n+1)= rac{x!}{(x-n)!}.

To distinguish the two, the notations x^{(n)} and (x)_n are commonly used to denote the rising and falling factorials, respectively. They are related by a difference in sign:

:(-x)^{(n)\mbox{ (rising)}} = (-1)^n (x)_n^{\mbox{(falling)}}

where (-1)^n is equal to +1 when ''n'' is even and −1 when ''n'' is odd.


ALTERNATE NOTATIONS

Another, less common notation was introduced by Ronald L. Graham , Donald E. Knuth and Oren Patashnik in their book '' Concrete Mathematics ''. They define, for the rising factorial:

:x^{\overline{n}}= rac{(x+n-1)!}{(x-1)!}

and for the falling factorial:

:x^{\underline{n}}= rac{x!}{(x-n)!}

Other notations for the falling factorial include ''P''(''x'', ''n''), ''x''P''n'', P''x'',''n'', or ''x''P''n''. (See Permutation and Combination ).

Another notation of the falling factorial using a function is:

: {Link without Title} ^{k/-h}=f(x)\cdot f(x-h)\cdot f(x-2h)\cdots f(x-(k+1)h)

where −''h'' is the decrement and ''k'' is the number of terms. The raising factorial is written:

: {Link without Title} ^{k/h}=f(x)\cdot f(x+h)\cdot f(x+2h)\cdots f(x+(k-1)h)

The Empty Product (''x'')0 is defined to be 1 in both cases.
Note that the falling factorial can be written as a Binomial Coefficient :

: rac{x^{\underline{n}}}{n!} = {x \choose n}

: rac{x^{ \overline{n}}}{n!} = {x+n-1 \choose n}

and thus a large number of identities on the binomial coefficients carry over to the Pochhammer symbols.

The rising factorial can be generalized to a continuous value of ''n''  using the Gamma Function :

:x^{\overline{n}}= rac{\Gamma(x+n)}{\Gamma(x)}

as can the falling factorial:

:x^{\underline{n}}= rac{\Gamma(x+1)}{\Gamma(x-n+1)}


RELATION TO UMBRAL CALCULUS


The falling factorial occurs in a formula which represents Polynomial s using the forward Difference Operator Δ and which is formally similar to Taylor's Theorem of Calculus . In this formula and in many other places, the falling factorial (''x'')''k'' in the calculus of Finite Difference s plays the role of ''x''''k'' in differential calculus. Note for instance the similarity of

:\Delta x^{\underline{k}} = k x^{\underline{k-1}}\,

and

:D x^k = k x^{k-1}\,

(where ''D'' denotes Differentiation with respect to ''x'').
The study of similarities of this type is known as Umbral Calculus . The general theory covering such relations, including the Pochhammer polynomials, is given by the theory of Polynomial Sequences Of Binomial Type and by Sheffer Sequence s.


NOTE

Pochhammer actually used (x)n to denote the binomial coefficient Knuth, ``Two notes on notation" .