generalized Ruiz’s identity

For $i,j=1,2,\mathrm{\dots }$, consider the polynomials $c_{i,j}(x)={(x+i)}^j-{(x+i-1)}^j$ ($x$ is the indeterminate). Then, for every positive natural number $n$,

$$det{(c_{i,j})}_{i,j=1}^n=\prod _{k=1}^{n}k!$$

Title	generalized Ruiz’s identity
Canonical name	GeneralizedRuizsIdentity
Date of creation	2013-03-22 14:31:45
Last modified on	2013-03-22 14:31:45
Owner	GeraW (6138)
Last modified by	GeraW (6138)
Numerical id	7
Author	GeraW (6138)
Entry type	Derivation
Classification	msc 11B65
Classification	msc 05A10
Related topic	determinant2
Related topic	Determinant2