Finding a matrix M whose characteristic polynomial gives orthogonal polynomials

[zetafunction]

given a set of orthogonal polynomials with respect to a certain measure w(x)

$$\int_{a}^{b}dx w(x) P_{n} (x)P_{m} (x) = \delta _{n,m}h_{n}$$

how can anybody prove that exists a certain M+M Hermitian matrix so

$$P_{m} (x)= < Det(1-xM)>$$ here <x> means average or expected value of 'x'

if we knew the set of orthogonal polynomials $$P_{m} (x)$$ for every 'm' and the measure w(x) , could we get the expression for the matrix M ??

 

[g_edgar]

Your equation doesn't make much sense to me. How about providing an example. A particular orthogonal system and the corresponding matrix.

 

[Dragonfall]

I recommend "proof by believe", i.e. write something that looks like a proof, believe it is a correct proof, without being able to check whether it is correct.

Looks like we got ourselves a troll.

 

[mathaino]

Interesting way to react to posts that do not tickle the own ears - delete them. You guys are not seekers of truth. Ibn al Haytham would be ashamed for you all.