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'''Petkovšek's algorithm''' is a [[computer algebra]] algorithm that computes a basis of [[hypergeometric identity|hypergeometric terms]] solution of its input linear recurrence equation with polynomial coefficients. Equivalently, it computes a first order right factor of linear [[difference operator]]s with polynomial coefficients.
== Examples ==
* Given the linear recurrence
:<math>4(n+2)(2n+3)(2n+5)a(n+2)
-12(2n+3)(9n^2+27n+22)a(n+1)
+81(n+1)( 3n+2)(3n+4) a(n) =0,</math>
the algorithm finds two linearly independent [[hypergeometric identity|hypergeometric terms]] that are solution:
:<math>{\frac {\Gamma \left( n+1 \right) }{\Gamma \left( n+3/2 \right) } \left( {\frac {27}{4}} \right) ^{n}},\qquad{\frac {\Gamma \left( n+2/3 \right) \Gamma \left( n+4/3 \right) }{\Gamma \left( n+3/2 \right) \Gamma \left( n+1 \right) } \left( {\frac {27}{4}} \right) ^{n}}.</math>
(Here, <math>\Gamma</math> denotes Euler's [[Gamma function]].) Note that the second solution is also a binomial coefficient <math>\binom{3n+1}{n}</math>, but it is not the aim of this algorithm to produce binomial expressions.
* Given the sum
:<math>a(n)=\sum_{k=0}^n{\binom{n}{k}^2\binom{n+k}{k}^2},</math>
coming from [[Roger Apéry|Apéry]]'s proof of the irrationality of <math>\zeta(3)</math>, [[Doron Zeilberger|Zeilberger]]'s algorithm computes the linear recurrence
:<math>(n+2)^3a(n+2)-(17n^2+51n+39)(2n+3)a(n+1)+(n+1)^3a(n)=0.</math>
Given this recurrence, the algorithm does not return any hypergeometric solution, which '''proves''' that <math>a(n)</math> does not simplify to a [[hypergeometric identity|hypergeometric term]].
== References ==
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