Polynomials arising in factoring generalized Vandermonde determinants: an algorithm for computing their coefficients

✍ Scribed by S. De Marchi

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 476 KB
• Volume: 34
• Category: Article
• ISSN: 0895-7177
• DOI: 10.1016/s0895-7177(01)00060-7

✦ Synopsis

We consider generalized Vandermonde determinants of the form vs;.(xl .... ,xs) = Ix~'~l, 1_< i, k < s,

where the x~ are distinct points belonging to an interval [a, b] of the real line, the index s stands for the order, the sequence # consists of ordered integers 0 <_ #1 < ~2 < • " < Ps. These determinants can be factored as a product of the classical Vandermonde determinant and a homogeneous symmetric function of the points involved, that is, a Schur function. On the other hand, we show that when

x --xs in the resulting polynomial, depending on the variable x, the Schur function can be factored as a

~1 times the (monic) polynomial 11i=1 t -xi), two-factors polynomial: the first is the constant 1-[~=11 x i while the second is a polynomial PM(X) of degree M ----ms-1 -s + 1.

Our main result is then the computation of the coefficients of the monic polynomial PM(X). We present an algorithm for the computation of the coefficients of Phi based on the Jacobi-Trudi identity for Schur functions.