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In combinatorics, Vandermonde's identity, or Vandermonde's convolution, named after Alexandre-Théophile Vandermonde (1772), states that : for binomial coefficients. This identity was given already in 1303 by the Chinese mathematician Zhu Shijie (Chu Shi-Chieh). See Askey 1975, pp. 59–60 for the history. There is a q-analog to this theorem called the q-Vandermonde identity. == Algebraic proof == In general, the product of two polynomials with degrees ''m'' and ''n'', respectively, is given by : 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, : Using the binomial theorem also for the exponents ''m'' and ''n'', and then the above formula for the product of polynomials, we obtain : where the above convention for the coefficients of the polynomials agrees with the definition of the binomial coefficients, because both give zero for all ''i'' > ''m'' and ''j'' > ''n'', respectively. By comparing coefficients of ''xr'', Vandermonde's identity follows for all integers ''r'' with 0 ≤ ''r'' ≤ ''m'' + ''n''. For larger integers ''r'', both sides of Vandermonde's identity are zero due to the definition of binomial coefficients. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Vandermonde's identity」の詳細全文を読む スポンサード リンク
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