In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in
hypergeometric series
In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
. These
identities occur frequently in solutions to
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
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 r ...
.
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
In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ...
.
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 ...
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 proofs have been found for many specific identities, there exist several general algorithms 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 can use to 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 the certificate to check the identity's proof:
* ''Definite sums'':
Sister Celine's Method,
Zeilberger Zeilberger () may refer to:
* Doron Zeilberger
Doron Zeilberger (; born 2 July 1950) is an Israeli-American mathematician, known for his work in combinatorics.
Education and career
He received his doctorate from the Weizmann Institute of Scien ...
's algorithm
* ''Indefinite sums'':
Gosper's algorithm
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
The book A = B by
Marko Petkovšek
Marko Petkovšek (1955 – 24 March 2023) was a Slovenian mathematician working mainly in symbolic computation.
He was a professor of discrete and computational mathematics at the University of Ljubljana. He is best known for Petkovšek's algorit ...
,
Herbert Wilf
Herbert Saul Wilf (June 13, 1931 – January 7, 2012) was an American mathematician, specializing in combinatorics and graph theory. He was the Thomas A. Scott Professorship of Mathematics, Thomas A. Scott Professor of Mathematics in Combinatori ...
and
Doron Zeilberger
Doron Zeilberger (; born 2 July 1950) is an Israeli-American mathematician, known for his work in combinatorics.
Education and career
He received his doctorate from the Weizmann Institute of Science in 1976, under the direction of Harry Dym, w ...
describes the three main approaches mentioned 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 rising factorial, fall ...
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