proof of generalized Ruiz’s identity


Consider the polynomialsMathworldPlanetmath ci,j(x)=(x+i)j-(x+i-1)j. Then, for every positive natural number n,


Consider the matrices M,C defined by Mi,j=(-1)j(i-1j-1) and Ci,j=(x+i)j-(x+i-1)j.

(MC)i,j =k=1n(-1)k(i-1k-1)((x+k)j-(x+k-1)j)

Therefore, by Ruiz’s identityPlanetmathPlanetmath, (MC)i,i=(-1)ii! for every i{1,,d} and (MC)i,j=0 for every i,j{1,,n} such that i>j. This means that MC is an upper triangular matrixMathworldPlanetmath whose main diagonal is -1!,2!,-3!,,(-1)nn!. Since the determinantMathworldPlanetmath of such a matrix is the product of the elements in the main diagonal, we get that detMC=(-1)nk=1nk!. It is easy to see that M itself is lower triangular with determinant (-1)n. Therefore detC=k=1nk!. ∎

Title proof of generalized Ruiz’s identity
Canonical name ProofOfGeneralizedRuizsIdentity
Date of creation 2013-03-22 14:32:02
Last modified on 2013-03-22 14:32:02
Owner GeraW (6138)
Last modified by GeraW (6138)
Numerical id 8
Author GeraW (6138)
Entry type Proof
Classification msc 11B65
Classification msc 05A10