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The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie. [1] There is a q-analog to this theorem called the q-Vandermonde identity. Vandermonde's identity can be generalized in numerous ways, including to the identity
As with the (non-q) Chu–Vandermonde identity, there are several possible proofs of the q-Vandermonde identity. The following proof uses the q -binomial theorem . One standard proof of the Chu–Vandermonde identity is to expand the product ( 1 + x ) m ( 1 + x ) n {\displaystyle (1+x)^{m}(1+x)^{n}} in two different ways.
Another way to derive the above formula is by taking a limit of the Vandermonde matrix as the 's approach each other. For example, to get the case of x 1 = x 2 {\displaystyle x_{1}=x_{2}} , take subtract the first row from second in the original Vandermonde matrix, and let x 2 → x 1 {\displaystyle x_{2}\to x_{1}} : this yields the ...
We can avoid writing large exponents for using the fact that for any exponent we have the identity =. This is the Vandermonde matrix for the roots of unity, up to the normalization factor. Note that the normalization factor in front of the sum ( 1 / N {\displaystyle 1/{\sqrt {N}}} ) and the sign of the exponent in ω are merely conventions, and ...
Euler's four-square identity; Euler's identity; Fibonacci's identity see Brahmagupta–Fibonacci identity or Cassini and Catalan identities; Heine's identity; Hermite's identity; Lagrange's identity; Lagrange's trigonometric identities; List of logarithmic identities; MacWilliams identity; Matrix determinant lemma; Newton's identity; Parseval's ...
which follows from Euler's integral formula by putting z = 1. It includes the Vandermonde identity as a special case. For the special case where =, (,;;) = () Dougall's formula generalizes this to the bilateral hypergeometric series at z = 1.
Vandermonde was a violinist, and became engaged with mathematics only around 1770. In Mémoire sur la résolution des équations (1771) he reported on symmetric functions and solution of cyclotomic polynomials ; this paper anticipated later Galois theory (see also abstract algebra for the role of Vandermonde in the genesis of group theory).
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