Search results
Results from the WOW.Com Content Network
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 ...
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.
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 determinant identity; Vandermonde's identity ...
In algebra, the Vandermonde polynomial of an ordered set of n variables , …,, named after Alexandre-Théophile Vandermonde, is the polynomial: = < (). (Some sources use the opposite order (), which changes the sign () times: thus in some dimensions the two formulas agree in sign, while in others they have opposite signs.)
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).
The name stems from the graphical representation of the identity on Pascal's triangle: when the addends represented in the summation and the sum itself are highlighted, the shape revealed is vaguely reminiscent of those objects (see hockey stick, Christmas stocking).