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Binomial Identity

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Roman (1984, p. 26) defines "the" binomial identity as the equation

p_n(x+y)=sum_(k=0)^n(n; k)p_k(y)p_(n-k)(x).
(1)

Iff the sequence p_n(x) satisfies this identity for all y in a field C of field characteristic 0, then p_n(x) is an associated sequence known as a binomial-type sequence.

In general, a binomial identity is a formula expressing products of factors as a sum over terms, each including a binomial coefficient (n; k). The prototypical example is the binomial theorem

(x+a)^n=sum_(k=0)^n(n; k)x^ka^(n-k)
(2)

for n>0. Abel (1826) gave a host of such identities (Riordan 1979, Roman 1984), some of which include

(x+y)(x+y-an)^(n-1) =sum_(k=0)^n(n; k)xy(x-ak)^(k-1)[y-a(n-k)]^(n-k-1),
(3)
x^(-1)(x+y-na)^n=sum_(k=0)^n(n; k)(x-ak)^(k-1)[y-a(n-k)]^(n-k)
(4)

(Abel 1826, Riordan 1979, p. 18; Roman 1984, pp. 30 and 73), and

x^(-1)(x+y)^n=sum_(k=0)^n(n; k)(x-ak)^(k-1)(y+ak)^(n-k)
(5)

(Saslaw 1989).

See also

Abel's Binomial Theorem, Abel Polynomial, Binomial, Binomial Coefficient, Dilcher's Formula, q-Abel's Theorem

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References

Abel, N. H. "Beweis eines Ausdrucks, von welchem die Binomial-Formel ein einzelner Fall ist." J. reine angew. Math. 1, 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. Combinatorics 3, No. 2, R16, 1, 1996. http://www.combinatorics.org/Volume_3/Abstracts/v3i2r16.html.Foata, D. "Enumerating k-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 Newsletter 10, 37-49, 1993.Strehl, V. "Binomial Identities--Combinatorial and Algorithmic Aspects." Discrete Math. 136, 309-346, 1994.

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Binomial Identity

Cite this as:

Weisstein, Eric W. "Binomial Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BinomialIdentity.html

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