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Determinant formulas

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A number of mathematical formulas can be written compactly using determinants. The following list contains some of the more useful or notable such formulas that have been discovered.

Extended quotient rule

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From the generalized product rule, if h=fg then

Using Cramer's rule to solve for f(n) produces the determinant formula [1]

By applying this to find Taylor series coefficients in the cases h=x, g=ex-1; h=ex-1, g=ex+1; h=sin x, g=cos x; h=x, g=sin x; and h=1, g=cos x; four different determinant expressions for the Bernoulli numbers and a determinant expression for the Euler numbers can be obtained.[2]

Symmetric polynomials

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The Schur polynomial

are defined as the quotients of the alternating polynomial

and the Vandermond determinant

This can, in turn, be expressed as a determinant involving the complete homogeneous symmetric polynomials as[3]

Newton's identities A002135 Number of terms in a symmetric determinant (See Muir p. 112)

  1. ^ J. W. L. Glaisher "Expression for Laplace's coefficients, Bernoullian and Eulerian numbers, &c, as Determinants" Messenger of Mathematics, vol. VI (1877), p. 60.
  2. ^ Muir, Thomas (1920). The Theory of Determinants in the Historical Order of Development. Vol. III. Macmillan and Co. p. 233.
  3. ^ Muir p. 135