Dear Rehouma Abdel Hamid ,

It is an elementary well known relation which is available in many undergraduate mathematics texts. In brief, assume that

p(x)=anxⁿ+an-1xⁿ⁻¹+.....a₁x+a₀ is a polynomial of degree n.

= an(x-λ₁)(x-λ₂)(x-λ₃).....(x-λn)

where ai (i=0,1,2,3,..n) are the coefficients of the polynomial p(x), an≠ 0,

and λi(i=1,2,3,...n) are the roots of the polynomial p(x)= 0.

If we have

s₁= (λ₁+λ₂+λ₃+...λn)= sum of the roots.

s₂ = (λ₁λ₂+λ₁λ₃+...λ₁λn+....λn-1λn)=

sum of the product of distinct roots taken 2 at a time.

etc,

sr=∑λi1λi2...λir= sum of the product of distinct roots taken r at a time.

sn= λ₁λ₂...λn

then the relation between the coefficients and the roots is given by:

sk=(-1)kan-k/an

For example:

s₁=(λ₁+λ₂+λ₃+...λn)= - an-1/an (k=1)

s₂=s₂ = (λ₁λ₂+λ₁λ₃+...λ₁λn+....λn-1λn)= an-2/an (k=2)

etc

sn = λ₁λ₂...λn = (-1)na0/an .

Best regards

Thank You

My interest is orthogonal polynomials in 2 complex variables.

( on contour of class of Geronimus).

it is pleasure to known others areas in polynomials studies.

Rehouma Abdel Hamid ,

The statement

< My interest is orthogonal polynomials in 2 complex variables.

( on contour of class of Geronimus).>

is not related to your original question.

Otherwise, you need to show what is in your mind!

You are welcome.

Thank You

it is pleasure

The test to prove it,is not return to original question but to help me to test these class of polynomials.

Rehouma Abdel Hamid ,

To test what?

Their roots and the relation with the coefficients? Or their orthogonality?

Thank you Dear Pr ISSAM

there are more test no ones we are to discover any answer to any question about associated orthogonal polynomials ?

my telephonne 00213554754978

Sincerely