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 ${\textstyle \mathbb {K} }$ be a field of characteristic zero. A nonzero sequence ${\textstyle y(n)}$ is called hypergeometric if the ratio of two consecutive terms is rational, i.e. ${\textstyle y(n+1)/y(n)\in \mathbb {K} (n)}$. The Petkovšek algorithm uses as key concept that this rational function has a specific representation, namely the Gosper-Petkovšek normal form. Let ${\textstyle r(n)\in \mathbb {K} [n]}$ be a nonzero rational function. Then there exist monic polynomials ${\textstyle a,b,c\in \mathbb {K} [n]}$ and ${\textstyle 0\neq z\in \mathbb {K} }$ such that

${\displaystyle r(n)=z{\frac {a(n)}{b(n)}}{\frac {c(n+1)}{c(n)}}}$

and

1. ${\textstyle \gcd(a(n),b(n+k))=1}$ for every nonnegative integer ${\textstyle k\in \mathbb {N} }$,
2. ${\textstyle \gcd(a(n),c(n))=1}$ and
3. ${\textstyle \gcd(b(n),c(n+1))=1}$.

This representation of ${\textstyle r(n)}$ 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 ${\textstyle c(n)}$. The other polynomials ${\textstyle a(n),b(n)}$ can be taken as the monic factors of the first coefficient polynomial ${\textstyle p_{0}(n)}$ resp. the last coefficient polynomial shifted ${\textstyle p_{r}(n-r+1)}$. Then ${\textstyle z}$ has to fulfill a certain algebraic equation. Taking all the possible finitely many triples ${\textstyle (a(n),b(n),z)}$ and computing the corresponding polynomial solution of the transformed recurrence equation ${\textstyle c(n)}$ gives a hypergeometric solution if one exists.^[1][3][4]

In the following pseudocode the degree of a polynomial ${\textstyle p(n)\in \mathbb {K} [n]}$ is denoted by ${\textstyle \deg(p(n))}$ and the coefficient of ${\textstyle n^{d}}$ is denoted by ${\textstyle {\text{coeff}}(p(n),n^{d})}$.

algorithm petkovsek is
    input: Linear recurrence equation ${\textstyle \sum _{k=0}^{r}p_{k}(n)\,y(n+k)=0,p_{k}\in \mathbb {K} [n],p_{0},p_{r}\neq 0}$.
    output: A hypergeometric solution ${\textstyle y}$ if there are any hypergeometric solutions.

    for each monic divisor ${\textstyle a(n)}$ of ${\textstyle p_{0}(n)}$ do
        for each monic divisor ${\textstyle b(n)}$ of ${\textstyle p_{r}(n-r+1)}$ do
            for each ${\textstyle k=0,\dots ,r}$ do
                ${\textstyle {\tilde {p}}_{k}(n)=p_{k}(n)\prod _{j=0}^{k-1}a(n+j)\prod _{i=k}^{r-1}b(n+i)}$
        ${\textstyle d=\max _{k=0,\dots ,r}\deg({\tilde {p}}_{k}(n))}$
        for each root ${\textstyle z}$ of ${\textstyle \sum _{k=0}^{r}{\text{coeff}}({\tilde {p}}_{k}(n),n^{d})z^{k}}$ do
            Find non-zero polynomial solution ${\textstyle c(n)}$ of ${\textstyle \sum _{k=0}^{r}z^{k}\,{\tilde {p}}_{k}(n)\,c(n+k)=0}$
            if such a non-zero solution ${\textstyle c(n)}$ exists then
                ${\textstyle r(n)=z\,a(n)/b(n)\,c(n+1)/c(n)}$
                return a non-zero solution ${\textstyle y(n)}$ of ${\textstyle y(n+1)=r(n)\,y(n)}$

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 ${\textstyle n\times n}$ can be described by the sequence ${\textstyle y(n)}$ which is determined by the recurrence equation

over ${\textstyle \mathbb {Q} }$. Taking ${\textstyle a(n)=n+1,b(n)=1}$ as monic divisors of ${\textstyle p_{0}(n)=4(n+1)^{2},p_{2}(n)=-1}$ respectively, one gets ${\textstyle z=\pm 2}$. For ${\textstyle z=2}$ the corresponding recurrence equation which is solved in Petkovšek's algorithm isThis recurrence equation has the polynomial solution ${\textstyle c(n)=c_{0}}$ for an arbitrary ${\textstyle c_{0}\in \mathbb {Q} }$. Hence ${\textstyle r(n)=2(n+1)}$ and ${\textstyle y(n)=2^{n}\,n!}$ 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 ${\displaystyle \zeta (3)}$

Given the sum

${\displaystyle a(n)=\sum _{k=0}^{n}{{\binom {n}{k}}^{2}{\binom {n+k}{k}}^{2}},}$

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

${\displaystyle (n+2)^{3}\,a(n+2)-(17n^{2}+51n+39)(2n+3)\,a(n+1)+(n+1)^{3}\,a(n)=0.}$

Given this recurrence, the algorithm does not return any hypergeometric solution, which proves that ${\displaystyle a(n)}$ does not simplify to a hypergeometric term.^[3]

References