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Chu–Vandermonde Identity
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients: :=\sum_^r for any nonnegative integers ''r'', ''m'', ''n''. The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie.See for the history. 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 : = \sum_ \cdots . Proofs Algebraic proof In general, the product of two polynomials with degrees ''m'' and ''n'', respectively, is given by :\biggl(\sum_^m a_ix^i\biggr) \biggl(\sum_^n b_jx^j\biggr) = \sum_^\biggl(\sum_^r a_k b_\biggr) x^r, where we use the convention that ''ai'' = 0 for all integers ''i'' > ''m'' and ''bj'' = 0 for all integers ''j'' > ''n''. By the binomial theorem, :(1+x)^ = \sum_^ x^r. U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |