A method for finding recurrence relations for hypergeometric polynomials directly from the series expansions of the polynomials. The method is effective and easily implemented, but usually slower than Zeilberger's algorithm. Given a sum , the method operates by finding a recurrence of the form
by proceeding as follows (Petkovšek et al. 1996, p. 59):
1. Fix trial values of and .
2. Assume a recurrence formula of the above form where are to be solved for.
3. Divide each term of the assumed recurrence by and reduce every ratio by simplifying the ratios of its constituent factorials so that only rational functions in and remain.
4. Put the resulting expression over a common denominator, then collect the numerator as a polynomial in .
5. Solve the system of linear equations that results after setting the coefficients of each power of in the numerator to 0 for the unknown coefficients .
6. If no solution results, start again with larger or .
Under suitable hypotheses, a "fundamental theorem" (Verbaten 1974, Wilf and Zeilberger 1992, Petkovšek et al. 1996) guarantees that this algorithm always succeeds for large enough and (which can be estimated in advance). The theorem also generalizes to multivariate sums and to - and multi--sums (Wilf and Zeilberger 1992, Petkovšek et al. 1996).