Factorisation, symmetric polynomials

[Kevin Smith]

suppose s_k are the coefficients obtained from expanding y(x)=prod(x_k-x), k=1,...,n, yielding the polynomial y(x)=sum_{k=0}^{n}s_kx^k. Then, does anybody have any clues as to factorising the polynomial sum_{k=0}^{n}a_ks_kx^k, for example, let x_k=choose(2k,k)? Is there some kind of n dimensional operator that expresses such a transformation?

 

[stapel]

suppose s_k are the coefficients obtained from expanding y(x)=prod(x_k-x), k=1,...,n, yielding the polynomial y(x)=sum_{k=0}^{n}s_kx^k. Then, does anybody have any clues as to factorising the polynomial sum_{k=0}^{n}a_ks_kx^k, for example, let x_k=choose(2k,k)? Is there some kind of n dimensional operator that expresses such a transformation?

Does the above mean the following?

Suppose s[sub:1luyn28y]k[/sub:1luyn28y] are the coefficients obtained from expanding the following product:

. . .\(\displaystyle \prod_{k=1}^n \, (x_k\, -\, x)\)

...which yields the following polynomial:

. . .\(\displaystyle y(x)\, =\, \sum_{k=0}^n \, s_k\, x^k\)

Does anybody have any clues regarding how to factor the following polynomial?

. . .\(\displaystyle \sum_{k=0}^n\, a_k\, s_k\, x^k\)

For example, would "let \(\displaystyle \begin{array}{c}\mbox{}\\ \scriptstyel x_k\, =\, {{2k}\choose{k}} \end{array}\) " be helpful at some stage? Is there some kind of n dimensional operator that expresses such a transformation?

Please reply with correction or confirmation. Thank you!

Eliz.

 

[Kevin Smith]

Thanks for the prompt reply Eliz!! Thats exactly what i mean, up to the last line where you write "would a_k=choose(2k,k) be helpful at some stage". I will explain a bit more..Let j \leq n be a positive integer. For a function y(x)=sum_{k=0}^{n}s_kx^k whose zeros are given, I am trying to factorize the expression Ay(x)=sum_{k=0}^{n}choose(2k,k-j)s_kx^k, where the s_k's are the elementary symmetric polynomials in the set of given zeros, i.e, {x_k}, k=1,...,n, that is- s_n=(-1)^n, s_{n-1}=(-1)^{n-1}(sum of all 1-tuples), s_{n-2}=(-1)^{n-2}(sum of all 2-tuples), and so on, down to s_0=x_1x_2...x_n. In other words, I am looking to find how the action of the operator A (which is defined as above) acts on the zeros of y, so I can factorize Ay (importantly, with zeros given as a function of the zeros of y).

I should let you know, this is part of a problem in Fourier analysis (sampling theory) which I have been working on for a long time, which leads to a method for finding closed forms of Riesz basis expansion formulae for a bandlimited functions, given a set of sample points (these are closely related to x_k's above). The above expression, which I have been grappling with for months, is the form of certain coefficients which arise in these expansion formula.

I have searched and searched the literature for something similar, but nobody seems to pose such a question (at least in any language that I understand), so I would be extemely gratefull for any suggestions you can offer..