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Synthesizing recurrence relations I: analysis of the problems

Published: 01 May 1983 Publication History

Abstract

Recurrence relations (RR) play an important role in many domains of Science. Mathematics, Physics or Computer Science are obvious examples. It is rather intriguing to remark that few attempts to handle them as generic objects have been made in the field of Computer Algebra [24]. The only genuine effort is due to Verbaeten [1] who studied recurrence relations of hypergeometric functions. The same type of functions was considered by Lafferty [2], who was interested in reducing them to elementary or special functions by doing pattern matching with RR.

References

[1]
P. Verbaeten, "The Automatic Construction of Pure Recurrence Relations", EUROSAM'74 Proceedings, ACM-SIGSAM Bulletin, 8, pp 96--98 (1974).
[2]
E. L. Lafferty, "Hypergeometric Function Reduction - An Adventure in Pattern Matching", Proceedings of the 1979 MACSYMA User's Conference, pp 465--481, Published by MIT Lab.
[3]
R. Caboz, Private communication.
[4]
A. Lonke and R. Caboz, "Stability Criterion for Hamiltonian Systems", Physica 99A, pp 350--356 (1979).
[5]
A. Pettrossi and R. M. Burstall, "Deriving Efficient Algorithms ..", Acta Informatica 18, pp 181--206 (1982).
[6]
G. S. Lueker, "Some Techniques for Solving Recurrence Relations", ACM. Comp. Surv. 12, pp 419--436 (1980).
[7]
W. Gautschi, "Computational Methods in Special Functions - A Survey", in "Theory and Applications of Special Functions". Ed. R. A. Askey, pp 1--98, Academic Press, N.Y. (1975).
[8]
N. J. A. Sloane, "A Handbook of Integer Sequences", Academic Press, N.Y. (1973).
[9]
D. Jarden, "Recurring Sequences", Riveon Lematematika, Jerusalem (1966).
[10]
G. Szegö, "Orthogonal polynomials", A.M.S. Colloquium Publication, vol. 23 (1939).
[11]
L. Ya. Geronimus, "Orthogonal Polynomials", Consultant Bureau, N.Y. (1961).
[12]
A. Endélyi et al., "Higher Transcendental Functions", Bateman Manuscript Project vol. I, II, McGraw-Hill, N.Y. (1953).
[13]
R. Askey, "Orthogonal Polynomials and Special Functions", Proceed. of Regional Conference Series in Applied Math. SIAM (1975).
[14]
E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis", 4th ed., Cambridge Univ. Press, Cambridge (1952).
[15]
D. Bessis et al., "Orthogonal polynomials on a family.", Letters in Math. Phys., 6, pp 123--140 (1982).
[16]
S. O. Rice, "Some properties of3F2(-n,n+1, Δ1, p;v)", Duke Math. J., 6, pp 108--119 (1940).
[17]
H. Bateman, "Two systems of polynomials for the solution of Laplace's integral equation", Duke Math. J., 2, pp 559--577 (1936).
[18]
L. J. Slater, "Generalized Hypergeometric functions", Cambridge Univ. Press, Cambridge, England (1966).
[19]
H. Buchholz, "The Confluent Hypergeometric function", Springer Trates in Natural Philosophy, vol. 15 (1969).
[20]
Sis. M. C. Fasenmyer, "A note on pure recurrence relations", Am. Math. Monthly, 56, pp 14--17 (1949).
[21]
H. T. Kung, "The Structure of parallel algorithms", Advances in Computers, vol. 19, pp 65--112 (1980).
[22]
E. Papon, "Algoritmes de detection de relations de récurrences.", Thèse 3ième cycle, Univ. Paris-Sud, Centre d'Orsay (1981) unpublished.
[23]
W. Küchlin, "Some Reduction Strategies for Algebraic Term Rewriting", ACM-SIGSAM Bulletin, 16(4), pp 13--23 (1982).
[24]
R. Loos, Introduction to "Computer Algebra", Ed. B. Buchberger et.al., Computing, Springer (1982).

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Published In

cover image ACM SIGSAM Bulletin
ACM SIGSAM Bulletin  Volume 17, Issue 2
May 1983
32 pages
ISSN:0163-5824
DOI:10.1145/1089330
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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 May 1983
Published in SIGSAM Volume 17, Issue 2

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