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Hypergeometric Function IdentitiesIn mathematics, hypergeometric identies are equalities involving sums over hypergeometric terms. These identities pop up very often in solutions to combinatorial problems and the analysis of algorithms. Although one used to prove these identities by hand, there exist now several algorithms to find and prove all hypergeometric identities. Examples -
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Definition There are two definitions of hypergeometric terms, both used in different cases as explained below. See also hypergeometric series. A term tk is a hypergeometric term if -
\frac{t_{k+1}}{t_k} is a rational function in k. A term F(n,k) is a hypergeometric term if -
\frac{F(n,k+1)}{F(n,k)} 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 beautiful proofs of certain identities there exist several 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 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: - Indefinite sums: Sister Celine's Method, Zeilberger's Algorithm
- Definite sums: Gosper's Algorithm
A book named A=B has been written by Marko Petkovsek, Herbert Wilf and Doron Zeilberger describing the three main approaches described above. External links
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