In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in
hypergeometric series. These
identities occur frequently in solutions to
combinatorial problems, and also in the
analysis of algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that re ...
.
These identities were traditionally found 'by hand'. There exist now several algorithms which can find and ''prove'' all hypergeometric identities.
Examples
:
:
:
:
Definition
There are two definitions of hypergeometric terms, both used in different cases as explained below. See also
hypergeometric series.
A term ''t
k'' is a hypergeometric term if
:
is a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
in ''k''.
A term ''F(n,k)'' is a hypergeometric term if
:
is a rational function in ''k''.
There exist two types of sums over hypergeometric terms, the definite and indefinite sums. A definite sum is of the form
:
The indefinite sum is of the form
:
Proofs
Although in the past one has found proofs of certain identities there exist several algorithms{{vague, date=December 2016 to find and prove identities. These algorithms first find a ''simple expression'' for a sum over hypergeometric terms and then provide a certificate which anyone could use to easily check and prove the correctness of the identity.
For each of the hypergeometric sum types there exist one or more methods to find a ''simple expression''. These methods also provide a certificate to easily check the proof of an identity:
* ''Definite sums'': Sister Celine's Method, Zeilberger's algorithm
* ''Indefinite sums'':
Gosper's algorithm
A book named A = B has been written by
Marko Petkovšek,
Herbert Wilf
Herbert Saul Wilf (June 13, 1931 – January 7, 2012) was a mathematician, specializing in combinatorics and graph theory. He was the Thomas A. Scott Professor of Mathematics in Combinatorial Analysis and Computing at the University of Pennsylv ...
and
Doron Zeilberger describing the three main approaches described above.
See also
*
Table of Newtonian series In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a_n written in the form
:f(s) = \sum_^\infty (-1)^n a_n = \sum_^\infty \frac a_n
where
:
is the binomial coefficient and (s)_n is the falling factorial. N ...
External links
The book "A = B" this book is freely downloadable from the internet.
Special-functions examplesat exampleproblems.com
Factorial and binomial topics
Hypergeometric functions
Mathematical identities
fr:Identités hypergéométriques