A hypergeometric series is a power series where the ratio of the coefficients of
xn+1 and xn is a
rational function of n.
The zeroes and poles of that function (usually assumed to be real numbers)
are considered explicit "parameters" in the "hypergeometric function"
defined as the sum of such a power series.
Many well-known functions are hypergeometric.
(2004-07-10) Upper factorial, or "rising factorial" (x)k A traditional notation, which may be ambiguous in other contexts.
In the theory of special functions, including hypergeometric functions,
the following notation is used for the "rising factorial", the product of k
increasing factors, one unit apart, starting with
a.
(x)k is called a Pochhammer symbol in honor of
Leo August Pochhammer (1841-1920) who introduced
generalized hypergeometric functions in 1870.
Competing Notations and Complementary Ones :
The notation introduced above is quite standard in the context of special
functions, but some authors have used it in combinatorics to denote the lower factorial
or falling factorial, which specifies a product like the above by giving
the highest of its factors instead.
In other words, they use (x)k for what we denote:
(x-k+1)k = k! C(x,k) = x k
The above leftmost notation for the falling factorial was introduced by
Ronald Graham,
Donald Knuth and
Oren Patashnik
(Concrete Mathematics,
1989).
For the sake of aesthetics,
they also proposed a symmetrical notation for the rising factorial
(using a superscript with an overbar)
which has not overtaken the more traditional notation we're using here.
The earliest notation for the "falling factorial" was [x]k .
It was introduced in 1772 by the founder of the Theory of
Determinants, the French mathematician Alexandre-Théophile
Vandermonde
(1735-1796) before there was even a standard notation for the "full" factorial
(now denoted n!) which Vandermonde would have denoted [n]n.
(2004-07-10) Gauss (1812) and Pochhammer (1870)
The hypergeometric function of Gauss was generalized in 1870.
In 1812, Gauss investigated the power series where the
coefficient of z k / k! is equal to
(u)k(v)k/ (w)k
for three constant parameters u,v,w.
F(u,v;w;z) =
åk
[ (u)k (v)k/ (w)k ] zk / k!
The generalization of this notation involves terms which have p components in their numerators
and q components in their denominators.
The numerator parameters come first (separated by commas) followed by a semicolon,
which separates them from the denominator parameters.
Another semicolon (sometimes a colon)
is used just before the function's argument, shown last.
It is customary to indicate the value of
p and q as subscripts before and after the symbol F.
These unnecessary subscripts improve readability,
but they are often dropped in the case of Gauss's original function.
( 2F1 = F). For example:
(2006-01-18) Sum of the reciprocals of Catalan numbers
(A121839)
An example of a sum expressible with the hypergeometric function:
1/1 + 1/1 + 1/2 + 1/5 + 1/14 + 1/42 + 1/132 + 1/429 + 1/1430 + ...
The reciprocals of Catalan numbers add up to
F ( 1 , 2 ; ½ ; ¼ ) namely:
