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© 2000-2018   Gérard P. Michon, Ph.D.


Confluent hypergeometric functions are to us as
trigonometric functions are to other people
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Hypergeometric function   |   Hypergeometric differential equation
Confluent hypergeometric function of the first kind (Kummer's function)
Confluent hypergeometric function of the second kind (Tricomi-Gordon)
Hypergeometric function  (MathWorld, by Eric W. Weisstein).
Wikipedia :   Hypergeometric Series

Hypergeometric Series, Hypergeometric Functions

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.

Loosely speaking, we define:   (a)k   =   (a+0) (a+1) (a+2) ... (a+k-1)

Precisely:   (a)0 = a  and  (a)k+1 = (a)k(a+k)   for any nonnegative integer k.

Alternately, in terms of the Gamma function:  (a)k   =   G(a+k) / G(a).

A few special cases are worth noting (see factorial and double-factorial):

  • (1)k   =   k!
  • (½)k   =   2-k (2k+1)!!
  • (-½)k   =   -2-k (2k-1)!!
  • (¼)k(-¼)k   =   16-k (4k-3)!!

(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].  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.

 Carl Friedrich Gauss 
 1777-1855 (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:

3F1 (u,v,w;x;z)   =     åk   [ (u)k (v)k (w)k / (x)k ]  zk / k!

Here are a few basic relations about  F =  2F1 :

  • F(u,v;w;z)   =   F(v,u;w,z)
  • F(u,v;v;z)   =   (1-z)-u
  • F(1,1;2,z)   =   ln(1-z) / z
  • dF/dz   =   F'(u,v;w;z)   =   (uv/w)  F(u+1,v+1;w+1;z)
  • Gauss's identity:
F(u,v;w,1)   =   G(w) G(w-u-v)
G(w-u) G(w-v)

(2004-07-10)   Kummer Transformations (1836)

These were first published by [Ernst] Eduard Kummer (1810-1893) in Crelle's Journal (Journal für die reine und angewandte Mathematik) Bd 15 (1836).

 Come back later, we're 
 still working on this one...

Exact Expressions for the Perimeter of an Ellipse

(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:

2 + 43/27   =   2.806133050770763489152923670063180325459584999+

Proof :

 Come back later, we're 
 still working on this one...

Sum of the reciprocal Catalan numbers  by  Juan Manuel Márquez Bobadilla  (CUCEI)

(2014-08-14)   Appell Series   (Paul Appell, 1880)
Four hypergeometric series of two variables   F1 ,  F2 ,  F3 ,  F4

These are two-variable generalizations of the one-variable hypergeometric series of Gauss  F = 2F1

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 still working on this one...

Appell series   |   Appell polynomials   |   Paul Appell (1855-1930)

(2016-06-05)   Zeilberger's Algorithm   (Doron Zeilberger, 1990)
WZ pairs.

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 still working on this one...

MathWorld :   Zeilberger's algorithm   |   Wilf-Zeilberger pairs
Wikipedia :   Gosper's algorithm   |   WZ-pairs
Herbert S. Wilf (1931-2012)   |   Bill Gosper (1943-)   |   Doron Zeilberger (1950-)

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