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Vandermonde Determinant

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Delta(x_1,...,x_n)=|1 x_1 x_1^2 ... x_1^(n-1); 1 x_2 x_2^2 ... x_2^(n-1); | | | ... |; 1 x_n x_n^2 ... x_n^(n-1)|
(1)
=product_(i,j; i>j)(x_i-x_j)
(2)

(Sharpe 1987). For integers a_1, ..., a_n, Delta(a_1,...,a_n) is divisible by product_(i=1)^(n)(i-1)! (Chapman 1996), the first few values of which are the superfactorials 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, ... (OEIS A000178).

See also

Superfactorial, Vandermonde Matrix

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References

Aldrovandi, R. Special Matrices of Mathematical Physics: Stochastic, Circulant and Bell Matrices. Singapore: World Scientific, p. 193, 2001.Chapman, R. "A Polynomial Taking Integer Values." Math. Mag. 69, 121, 1996.Fletcher, A.; Miller, J. C. P.; Rosenhead, L.; and Comrie, L. J. An Index of Mathematical Tables, Vol. 1. Reading, MA: Addison-Wesley, p. 50, 1962.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1111, 2000.Graham, R. L.; Knuth, D. E.; and Patashnik, O. "Binomial Coefficients." Ch. 5 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 231, 1994.Radoux, C. "Query 145." Not. Amer. Math. Soc. 25, 197, 1978.Ryser, H. J. Combinatorial Mathematics. Buffalo, NY: Math. Assoc. Amer., p. 53, 1963.Sharpe, D. §2.9 in Rings and Factorization. Cambridge, England: Cambridge University Press, 1987.Sloane, N. J. A. Sequence A000178/M2049 in "The On-Line Encyclopedia of Integer Sequences."

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Vandermonde Determinant

Cite this as:

Weisstein, Eric W. "Vandermonde Determinant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VandermondeDeterminant.html

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