Sister Celine's Method
A method for finding Recurrence Relations for hypergeometric polynomials directly from theseries expansions of the polynomials. The method is effective and easily implemented, but usually slower thanZeilberger's Algorithm. Given a sum 
, the method operates by finding a recurrence of the form
by proceeding as follows (Petkovsek 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 simplifyingthe 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 .
and (which can be estimated in advance). The theorem also generalizesto multivariate sums and to 
- and multi--sums (Wilf and Zeilberger 1992, Petkovsek et al. 1996).See also Generalized Hypergeometric Function, Gosper's Algorithm, Hypergeometric Identity,Hypergeometric Series, Zeilberger's Algorithm
References
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. Monthly 56, 14-17, 1949. Petkovsek, 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. 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.
- Algorithmic Complexity
- Alhazen's Billiard Problem
- Alhazen's Problem
- Aliasing
- Aliquant Divisor
- Aliquot Cycle
- Aliquot Divisor
- Aliquot Sequence
- Alladi-Grinstead Constant
- Allais Paradox
- Allegory
- Allometric
- All-Poles Model
- Almost All
- Almost Alternating Knot
- Almost Alternating Link
- Almost Everywhere
- Almost Integer
- Almost Perfect Number
- Almost Prime
- Alpha
- Alphabet
- Alpha Function
- Alphamagic Square
- Alphametic