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).

Fasenmyer, Sister M. C. Some Generalized Hypergeometric Polynomials. Ph.D. thesis. University of Michigan, Nov. 1945.Fasenmyer,
Sister M. C. "Some Generalized Hypergeometric Polynomials." Bull.
Amer. Math. Soc.53, 806-812, 1947.Fasenmyer, Sister M. C.
"A Note on Pure Recurrence Relations." Amer. Math. Monthly56,
14-17, 1949.Koepf, W. "Holonomic Recurrence Equations." Ch. 4
in Hypergeometric
Summation: An Algorithmic Approach to Summation and Special Function Identities.
Braunschweig, Germany: Vieweg, pp. 44-60, 1998.Petkovšek,
M.; Wilf, H. S.; and Zeilberger, D. "Sister Celine's Method." Ch. 4
in A=B.
Wellesley, MA: A K Peters, pp. 55-72, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.Rainville,
E. D. Chs. 14 and 18 in Special
Functions. New York: Chelsea, 1971.Verbaten, P. "The Automatic
Construction of Pure Recurrence Relations." Proc. EUROSAM '74, ACM-SIGSAM
Bull.8, 96-98, 1974.Wilf, H. S. and Zeilberger, D.
"An Algorithmic Proof Theory for Hypergeometric (Ordinary and "") Multisum/Integral Identities." Invent. Math.108,
575-633, 1992.