Exercise 24, § 5

We take up again the notations of the preceding exercises; in particular, we consider a sequence $\boldsymbol{\mu}= (\mu_n)_{n\in\mathbf{N}}$ of real numbers satisfying conditions (24) and such that the sesquilinear form $b_{\boldsymbol{\mu}}$ is separating.

a) There exists a unique sequence of polynomials $(q_n)_{n\in\mathbf{N}}$ with real coefficients such that for every $s\in \mathbf{C}$, the formula

is valid ("orthogonal polynomials of the second species").

b) Let $\nu \in \mathscr{M}_m(\mathbf{R})$ be a measure on $\mathbf{R}$ such that $\nu (x^n) =\mu_n$ for every $n\in \mathbf{N}$. For every $s\in \mathbf{C}-\mathbf{R}$, one has

c) Let $s\in$ C-R. The sequence $(p_n(s))_{n\in\mathbf{N}}$ belongs to $\ell^2(\mathbf{N})$ if and only if the sequence $(q_n(s))_{n\in\mathbf{N}}$ belongs to $\ell^2(\mathbf{N})$. (Let $\sigma (s) =\int(t-s)^{-1}d\nu (t)$; prove with the aid of the preceding question that the sequence $(q_n(s) +\sigma (s)p_n(s))_{n\in\mathbf{N}}$ belongs to $\ell^2(\mathbf{N}).$)

d) For every $n\in \mathbf{N}$, the formula

is valid in $\mathbf{C}[X,Y]$.

e) For every $n\in \mathbf{N}$, one has the relation

f) If the series with general term $a^{-1}_n$ is divergent, then the measure $\nu$ is the unique measure such that $\nu (x^n) =\mu_n$ for every $n\in \mathbf{N}$.