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In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all i > m and j > n, respectively. By comparing coefficients of x r , Vandermonde's identity follows for all integers r with 0 ≤ r ≤ m + n .
The following proof uses the q-binomial theorem. One standard proof of the Chu–Vandermonde identity is to expand the product (+) (+) in two different ways. Following Stanley, [1] we can tweak this proof to prove the q-Vandermonde identity, as well. First, observe that the product
As a 16-year-old, Abel gave a rigorous proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals. [13] [14] Abel wrote a fundamental work on the theory of elliptic integrals, containing the foundations of the theory of elliptic functions.
Marden's theorem. This theorem relating the location of the zeros of a complex cubic polynomial to the zeros of its derivative was named by Dan Kalman after Kalman read it in a 1966 book by Morris Marden, who had first written about it in 1945. [8] But, as Marden had himself written, its original proof was by Jörg Siebeck in 1864. [9]
In mathematics, Bertrand's postulate (now a theorem) states that, for each , there is a prime such that < <.First conjectured in 1845 by Joseph Bertrand, [1] it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan.
In 370 BC, Plato's Parmenides may have contained traces of an early example of an implicit inductive proof, [5] however, the earliest implicit proof by mathematical induction was written by al-Karaji around 1000 AD, who applied it to arithmetic sequences to prove the binomial theorem and properties of Pascal's triangle.
In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum.