In general, a binomial identity is a formula expressing products of factors as a sum over terms, each including a binomial coefficient . The prototypical example is the
binomial theorem
(2)
for .
Abel (1826) gave a host of such identities (Riordan 1979, Roman 1984), some of which
include
(3)
(4)
(Abel 1826, Riordan 1979, p. 18; Roman 1984, pp. 30 and 73), and
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159-160, 1826. Reprinted in Œuvres Complètes, 2nd ed., Vol. 1.
pp. 102-103, 1881.Bhatnagar, G. Inverse Relations, Generalized
Bibasic Series, and their U(n) Extensions. Ph.D. thesis. Ohio State University,
p. 61, 1995.Comtet, L. Advanced
Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht,
Netherlands: Reidel, p. 128, 1974.Ekhad, S. B. and Majewicz,
J. E. "A Short WZ-Style Proof of Abel's Identity." Electronic J.
Combinatorics3, No. 2, R16, 1, 1996. http://www.combinatorics.org/Volume_3/Abstracts/v3i2r16.html.Foata,
D. "Enumerating -Trees."
Discr. Math.1, 181-186, 1971.Riordan, J. Combinatorial
Identities. New York: Wiley, p. 18, 1979.Roman, S. "The
Abel Polynomials." §4.1.5 in The
Umbral Calculus. New York: Academic Press, pp. 29-30 and 72-75, 1984.Saslaw,
W. C. "Some Properties of a Statistical Distribution Function for Galaxy
Clustering." Astrophys. J.341, 588-598, 1989.Strehl,
V. "Binomial Sums and Identities." Maple Technical Newsletter10,
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Algorithmic Aspects." Discrete Math.136, 309-346, 1994.