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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
The hockey stick identity confirms, for example: for n=6, r=2: 1+3+6+10+15=35. In combinatorics , the hockey-stick identity , [ 1 ] Christmas stocking identity , [ 2 ] boomerang identity , Fermat's identity or Chu's Theorem , [ 3 ] states that if n ≥ r ≥ 0 {\displaystyle n\geq r\geq 0} are integers, then
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.
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.
Visual proof of the Pythagorean identity: for any angle , the point (,) = (, ) lies on the unit circle, which satisfies the equation + =.Thus, + =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...
For example, quotient set, quotient group, quotient category, etc. 3. In number theory and field theory, / denotes a field extension, where F is an extension field of the field E. 4. In probability theory, denotes a conditional probability. For example, (/) denotes the probability of A, given that B occurs.
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 ...
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 ...