Determining a measure based on recurrence relation for orthogonal polynomials Suppose you were handed a sequence of polynomials $P_n(x)$ such that the $n$th one is degree $n$ and such that they satisfy a constant coefficient recurrence relation whose coefficients are taken from $\mathbb{C}[x]$. A classic example of this is given by the Chebyshev polynomials (of the first kind) which satisfy the recurrence relation:
$$P_{n+1}(x) = 2x P_{n}(x) - P_{n-1}(x).$$
As is well known, these polynomials are orthogonal with respect to the measure $\frac{1}{\sqrt{1-x^2}} dx$ on $[-1, 1]$.
My question, is how, given the recurrence relation (and initial conditions), one could determine a (the unique?) measure with respect to which these polynomials are orthogonal. I am particularly interested in the case of the form $P_{n+1}(x) = a x P_{n}(x) + b P_{n-1}(x)$, but am also interested in the more general theory.
I understand that there is a well developed machinery for these sorts of questions, but the literature on this subject often does things in too great of generality for me to follow how it applies in the simplest of cases (two term constant coefficient recurrence relations). I am very much a novice to these sorts of questions. I would very much appreciate an elementary reference for this subject, or if someone could explain how it works in the Chebyshev polynomial case (i.e. determining the measure without prior knowledge of the measure).
 A: If you know the recurrence relation you can write down the Jacobi operator $J$ describing the matrix by which multiplication by $x$ acts on the basis $P_n(x)$. Then there is a formula for the moments
$$m_n = \int x^n \, d \mu$$
of the unknown measure in terms of the coefficients of $J$, which says that $m_n = (J^n)_{1,1}$ (the first entry in the first row), at least if $P_0(x) = 1$. You can see the first few such formulas in this blog post (where I don't name the Jacobi operator because I didn't know it was called that).
In your case $J$ turns out to be particularly simple. The recurrence relation determines the action of multiplication by $x$ via
$$x P_n(x) = \frac{P_{n+1}(x) + P_{n-1}(x)}{2}$$
although because the initial conditions are $P_0(x) = 1, P_1(x) = x$, this doesn't hold when $n = 0$ unless we define $P_{-1}(x) = x$; this is entirely consistent with the identity $P_n(\cos \theta) = \cos n \theta$ and it turns out to make $J$ look nicer and easier to analyze. In fact extending the recurrence above to negative values of $n$ by letting $P_{-n}(x) = P_n(x)$ makes $J$ a "$\mathbb{Z} \times \mathbb{Z}$" infinite matrix
$$\displaystyle J = \left[ \begin{array}{cccccc}
& \vdots & \vdots & \vdots & \vdots &  \\
\cdots &0 & \frac{1}{2} & 0 & 0 & \cdots \\
\cdots & \frac{1}{2} & 0 & \frac{1}{2} & 0 & \cdots \\
\cdots & 0 & \frac{1}{2} & 0 & \frac{1}{2} & \cdots \\
\cdots & 0 & 0 & \frac{1}{2} & 0 & \cdots \\
 & \vdots & \vdots & \vdots & \vdots & \\ \end{array} \right].$$
Then $(J^n)_{11}$ can be described combinatorially as $\frac{1}{2^n}$ times the number of paths on the infinite graph $C_{\infty} = \{ \dots -1 \leftrightarrow 0 \leftrightarrow 1 \leftrightarrow \dots \}$ which begin and end at the "origin." This is $0$ if $n$ is odd and ${n \choose n/2}$ if $n$ is even, which gives our moment sequence
$$m_n = \begin{cases} 0 & \text{ if } n \text{ odd} \\
 \frac{1}{2^n} {n \choose n/2} & \text{ if } n \text{ even} \end{cases}.$$
Conversely given these moments it's not hard to reproduce the recurrence relation (I describe how in the linked post) so the data of one is more or less equivalent to the data of the other. At this point we are reduced to solving the Hamburger moment problem. You may or may not recognize this sequence as the sequence
$$m_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} \cos^n \theta \, d \theta$$
(write $\cos \theta = \frac{e^{i \theta} + e^{-i \theta}}{2}$) but if you did you'd be basically done. A final change of coordinates $x = \cos \theta, dx = -\sin \theta \, d \theta = \pm \sqrt{1 - x^2} \, d \theta$ gives
$$m_n = \frac{1}{\pi} \int_{-1}^1 x^n \frac{dx}{\sqrt{1 - x^2}}.$$
An alternative approach that might work a bit more generally is to use the moments to compute the moment generating function resp. the characteristic function of the measure and then see if you can either recognize that or invert the Laplace resp. Fourier transform, but personally I don't recognize either in this case.
