Petkovšek's algorithm

Petkovšek's algorithm is a computer algebra algorithm that computes a basis of hypergeometric terms solution of its input linear recurrence equation with polynomial coefficients. Equivalently, it computes a first order right factor of linear difference operators with polynomial coefficients. This algorithm is implemented in all the major computer algebra systems.

the algorithm finds two linearly independent hypergeometric terms that are solution:

(Here, ${\displaystyle \Gamma }$ denotes Euler's Gamma function.) Note that the second solution is also a binomial coefficient ${\displaystyle {\binom {3n+1}{n}}}$, but it is not the aim of this algorithm to produce binomial expressions.

coming from Apéry's proof of the irrationality of ${\displaystyle \zeta (3)}$, Zeilberger's algorithm computes the linear recurrence

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