Petkovšek's algorithm

Petkovšek's algorithm (also Hyper) 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 was developed by Marko Petkovšek in his PhD-thesis 1992.[1] The algorithm is implemented in all the major computer algebra systems.

Gosper-Petkovšek representation

Let be a field of characteristic zero. A nonzero sequence is called hypergeometric if the ratio of two consecutive terms is rational, i.e. . The Petkovšek algorithm uses as key concept that this rational function has a specific representation, namely the Gosper-Petkovšek normal form. Let be a nonzero rational function. Then there exist monic polynomials and such that

and

  1. for every nonnegative integer ,
  2. and
  3. .

This representation of is called Gosper-Petkovšek normal form. These polynomials can be computed explicitly. This construction of the representation is an essential part of Gosper's algorithm.[2] Petkovšek added the conditions 2. and 3. of this representation which makes this normal form unique.[1]

Algorithm

Using the Gosper-Petkovšek representation one can transform the original recurrence equation into a recurrence equation for a polynomial sequence . The other polynomials can be taken as the monic factors of the first coefficient polynomial resp. the last coefficient polynomial shifted . Then has to fulfill a certain algebraic equation. Taking all the possible finitely many triples and computing the corresponding polynomial solution of the transformed recurrence equation gives a hypergeometric solution if one exists.[1][3][4]

In the following pseudocode the degree of a polynomial is denoted by and the coefficient of is denoted by .

algorithm petkovsek is
    input: Linear recurrence equation .
    output: A hypergeometric solution  if there are any hypergeometric solutions.

    for each monic divisor  of  do
        for each monic divisor  of  do
            for each  do
                
        
        for each root  of  do
            Find non-zero polynomial solution  of 
            if such a non-zero solution  exists then
                
                return a non-zero solution  of 

If one does not end if a solution is found it is possible to combine all hypergeometric solutions to get a general hypergeometric solution of the recurrence equation, i.e. a generating set for the kernel of the recurrence equation in the linear span of hypergeometric sequences.[1]

Petkovšek also showed how the inhomogeneous problem can be solved. He considered the case where the right-hand side of the recurrence equation is a sum of hypergeometric sequences. After grouping together certain hypergeometric sequences of the right-hand side, for each of those groups a certain recurrence equation is solved for a rational solution. These rational solutions can be combined to get a particular solution of the inhomogeneous equation. Together with the general solution of the homogeneous problem this gives the general solution of the inhomogeneous problem.[1]

Examples

Signed permutation matrices

The number of signed permutation matrices of size can be described by the sequence which is determined by the recurrence equation

over . Taking as monic divisors of respectively, one gets . For the corresponding recurrence equation which is solved in Petkovšek's algorithm is

This recurrence equation has the polynomial solution for an arbitrary . Hence and is a hypergeometric solution. In fact it is (up to a constant) the only hypergeometric solution and describes the number of signed permutation matrices.[5]

Irrationality of

Given the sum

coming from Apéry's proof of the irrationality of , Zeilberger's algorithm computes the linear recurrence

Given this recurrence, the algorithm does not return any hypergeometric solution, which proves that does not simplify to a hypergeometric term.[3]

References

  1. Petkovšek, Marko (1992). "Hypergeometric solutions of linear recurrences with polynomial coefficients". Journal of Symbolic Computation. 14 (2–3): 243–264. doi:10.1016/0747-7171(92)90038-6. ISSN 0747-7171.
  2. Gosper, R. William (1978). "Decision procedure for indefinite hypergeometric summation" (PDF). Proc. Natl. Acad. Sci. USA. 75 (1): 40–42. doi:10.1073/pnas.75.1.40. PMC 411178. PMID 16592483.
  3. Petkovšek, Marko; Wilf, Herbert S.; Zeilberger, Doron (1996). A=B. A K Peters. ISBN 1568810636. OCLC 33898705.
  4. Kauers, Manuel; Paule, Peter (2011). The concrete tetrahedron : symbolic sums, recurrence equations, generating functions, asymptotic estimates. Wien: Springer. ISBN 9783709104453. OCLC 701369215.
  5. "A000165 - OEIS". oeis.org. Retrieved 2018-07-02.
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