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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.
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 identity; Pfister's sixteen-square identity; Sherman–Morrison formula; Sophie Germain identity; Sun's curious identity; Sylvester's ...
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 ...
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.
Download as PDF; Printable version; ... It is a generalization of Vandermonde's identity, ... "Two matrix inversions associated with the Hagen-Rothe formula, ...
The sodium ion is an example of a cationic species and its formula is Na +. Species with an overall negative charge will be an anionic species. Chloride is an anionic species, and its formula is Cl −. [7] Radical species: Molecules or atoms with unpaired electrons. Triarlborane anion is a radical species and its formula is Ar3B − [8] [9]
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).