Moments, Hankel determinants, orthogonal polynomials, Motzkin paths, and continued fractions

Previously we described all finite-dimensional random algebras with faithful states. In this post we will describe states on the infinite-dimensional Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ . Along the way we will run into and connect some beautiful and classical mathematical objects.

A special case of part of the following discussion can be found in an old post on the Catalan numbers.

Positivity and Hankel determinants

Consider the Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ with involution given by extending Image may be NSFW.
Clik here to view. $x^{\dagger} = x$ ; in other words, the free complex Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebra on a self-adjoint element. A state Image may be NSFW.
Clik here to view. $\mathbb{E} : \mathbb{C}[x] \to \mathbb{C}$ is uniquely determined by the moments Image may be NSFW.
Clik here to view. $m_n = \mathbb{E}(x^n)$ which are real since the Image may be NSFW.
Clik here to view. x^n are all self-adjoint and which satisfy Image may be NSFW.
Clik here to view. m_0 = 1 . Positivity is equivalent to the condition that for any Image may be NSFW.
Clik here to view. $c(x) = c_0 + c_1 x + ... + c_n x^n \in \mathbb{C}[x]$ we have

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(c(x)^{\dagger} c(x)) = \sum_{i, j} \overline{c_i} c_j m_{i+j} \ge 0$

and this condition characterizes states on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ . That this condition actually characterizes Borel measures on the real line Image may be NSFW.
Clik here to view. $\mathbb{R}$ is the content of the solution to the Hamburger moment problem, although we will not use this fact. In discussing examples, we will make implicit use of the fact that various kinds of Borel measures on Image may be NSFW.
Clik here to view. $\mathbb{R}$ are uniquely determined by their moments thanks to results like Carleman’s condition, but only in order to identify these measures from their moments.

Note that by the universal property, if Image may be NSFW.
Clik here to view. is any random algebra and Image may be NSFW.
Clik here to view. $a \in A$ any self-adjoint element, then there is a unique morphism Image may be NSFW.
Clik here to view. $\mathbb{C}[x] \to A$ of Image may be NSFW.
Clik here to view. $^{\dagger}$ -algebras sending Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view., and pulling back the state on Image may be NSFW.
Clik here to view. gives a state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ . Consequently, everything we are about to say about states on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ places restrictions on states on any random algebra (more precisely, on moment sequences of self-adjoint elements of any random algebra).

The states which are not faithful are straightforward to describe.

Proposition: Let Image may be NSFW.
Clik here to view. $\mathbb{E}$ be a state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ which is not faithful. If Image may be NSFW.
Clik here to view. is a nonzero self-adjoint element of minimal degree such that Image may be NSFW.
Clik here to view. $\mathbb{E}(c^2) = 0$ , then Image may be NSFW.
Clik here to view. $\mathbb{E}$ is a finite sum of Dirac measures supported at the roots of Image may be NSFW.
Clik here to view., all of which are real.

Proof. Note that “self-adjoint” here is equivalent to having real coefficients. We first show that the roots of Image may be NSFW.
Clik here to view. are all real. Suppose by contradiction that Image may be NSFW.
Clik here to view. c = c(x) has a complex root Image may be NSFW.
Clik here to view. r + si with Image may be NSFW.
Clik here to view. $s \neq 0$ . Then it is divisible by Image may be NSFW.
Clik here to view. (x - r - si)(x - r + si) . Writing Image may be NSFW.
Clik here to view. c(x) = d(x) ((x - r)^2 + s^2) where Image may be NSFW.
Clik here to view. is self-adjoint, we have

Image may be NSFW.
Clik here to view. $0 = \mathbb{E}(c^2) \ge \mathbb{E}(d^2 s^4)$ .

By positivity of Image may be NSFW.
Clik here to view. $\mathbb{E}$ , it follows that Image may be NSFW.
Clik here to view. $\mathbb{E}(d^2 s^4) = 0$ , and since Image may be NSFW.
Clik here to view. s^4 > 0 , it follows that Image may be NSFW.
Clik here to view. $\mathbb{E}(d^2) = 0$ , which contradicts the assumption that Image may be NSFW.
Clik here to view. has minimal degree. Hence all of the roots of Image may be NSFW.
Clik here to view. are real.

By the division algorithm, we can write any Image may be NSFW.
Clik here to view. $a \in \mathbb{C}[x]$ in the form Image may be NSFW.
Clik here to view. a = cq + r where Image may be NSFW.
Clik here to view. $\deg r < \deg c$ . By Cauchy-Schwarz we have Image may be NSFW.
Clik here to view. $\mathbb{E}(cq) = 0$ , hence

Image may be NSFW.
Clik here to view. $\mathbb{E}(a) = \mathbb{E}(cq + r) = \mathbb{E}(r)$ .

Let Image may be NSFW.
Clik here to view. x_1, ... x_n be the real roots of Image may be NSFW.
Clik here to view.. Since Image may be NSFW.
Clik here to view. $\deg r < \deg c$ , by Lagrange interpolation we know that Image may be NSFW.
Clik here to view. can be written

Image may be NSFW.
Clik here to view. $\displaystyle r(x) = \sum_{k=1}^n r(x_i) \ell_i(x)$

where

Image may be NSFW.
Clik here to view. $\displaystyle \ell_i(x) = \prod_{1 \le m \le n, m \neq i} \frac{x - x_m}{x_i - x_m}$

are the Lagrange interpolation polynomials, which do not depend on Image may be NSFW.
Clik here to view.. Consequently,

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(a) = \mathbb{E}(r) = \sum_{k=1}^n r(x_i) \mathbb{E}(\ell_i) = \sum_{k=1}^n a(x_i) \mathbb{E}(\ell_i)$

which is a finite sum of Dirac measures supported at the points Image may be NSFW.
Clik here to view. x_i as desired. Image may be NSFW.
Clik here to view. $\Box$

The corresponding universal statement is the following: if Image may be NSFW.
Clik here to view. is a self-adjoint element of any random algebra Image may be NSFW.
Clik here to view. which has a minimal polynomial, then the state on Image may be NSFW.
Clik here to view. restricted to Image may be NSFW.
Clik here to view. $\mathbb{C}[a]$ is a finite sum of Dirac measures supported on the spectrum of Image may be NSFW.
Clik here to view. (which is finite).

As for the faithful states, we can say the following. Faithfulness is equivalent to the condition that for every Image may be NSFW.
Clik here to view. the Hankel matrix

Image may be NSFW.
Clik here to view. $\displaystyle H_n = \left[ \begin{array}{cccc} m_0 & m_1 & \hdots & m_{n-1} \\ m_1 & m_2 & \hdots & m_n \\ \vdots & \vdots & \ddots & \vdots \\ m_{n-1} & m_n & \hdots & m_{2n-2} \end{array} \right]$

with entries Image may be NSFW.
Clik here to view. $(H_n)_{i, j} = m_{i+j}$ is positive-definite. This is because Image may be NSFW.
Clik here to view. H_n is the symmetric matrix describing the restriction of the inner product Image may be NSFW.
Clik here to view. $\langle a, b \rangle = \mathbb{E}(a^{\dagger} b)$ to the subspace Image may be NSFW.
Clik here to view. V_n of polynomials of degree less than Image may be NSFW.
Clik here to view.. In particular, if Image may be NSFW.
Clik here to view. $\mathbb{E}$ is faithful, the Hankel determinants Image may be NSFW.
Clik here to view. $h_n = \det H_n$ are positive. For example, Image may be NSFW.
Clik here to view. h_2 = m_2 - m_1^2 is the variance Image may be NSFW.
Clik here to view. $\text{Var}(x) = \mathbb{E}(x^2) - \mathbb{E}(x)^2$ .

Proposition (Sylvester’s criterion): Image may be NSFW.
Clik here to view. H_n is positive-definite if and only if Image may be NSFW.
Clik here to view. h_k > 0 for all Image may be NSFW.
Clik here to view. $k \le n$ .

Corollary: A moment sequence Image may be NSFW.
Clik here to view. m_n determines a faithful state if and only if Image may be NSFW.
Clik here to view. h_n > 0 for all Image may be NSFW.
Clik here to view..

Proof. We observed one direction above. In the other direction, we prove the contrapositive. Note that Image may be NSFW.
Clik here to view. H_1 is always positive-definite. We proceed by induction. Assume that Image may be NSFW.
Clik here to view. $H_{n-1}$ is positive-definite. By the spectral theorem, Image may be NSFW.
Clik here to view. H_n has an orthonormal basis of eigenvectors, which we may take to be self-adjoint elements of Image may be NSFW.
Clik here to view. V_n . Since Image may be NSFW.
Clik here to view. h_n > 0 , it follows that Image may be NSFW.
Clik here to view. H_n has an even number of eigenvectors with negative eigenvalues. Suppose by contradiction that it has at least one, hence at least two, such eigenvectors Image may be NSFW.
Clik here to view. v_i, v_j with negative eigenvalues Image may be NSFW.
Clik here to view. $\lambda_i, \lambda_j$ . Then

Image may be NSFW.
Clik here to view. $\displaystyle (a v_i + b v_j)^T H_n (a v_i + b v_j) = \lambda_i a^2 + \lambda_j b^2 \le 0$

with strict inequality if either Image may be NSFW.
Clik here to view. or Image may be NSFW.
Clik here to view. is nonzero. On the other hand, there exists some choice of Image may be NSFW.
Clik here to view. a, b such that Image may be NSFW.
Clik here to view. $a v_i + b v_j \in V_{n-1}$ , from which it follows that Image may be NSFW.
Clik here to view. $H_{n-1}$ is not positive-definite; contradiction.

Hence Image may be NSFW.
Clik here to view. H_n has only positive eigenvalues, so is positive-definite. Image may be NSFW.
Clik here to view. $\Box$

So we have reduced the problem of determining whether a moment sequence describes a state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ to the problem of determining whether its Hankel determinants are non-negative. Unfortunately, it is not at all obvious how to compute Hankel determinants. We give without proof several evaluations of Hankel determinants below; the proofs will be subsumed in a more general result later in the post.

Example. Consider the moment sequence Image may be NSFW.
Clik here to view. $m_n = \frac{1}{n+1}$ . The corresponding Hankel matrices are the Hilbert matrices

Image may be NSFW.
Clik here to view. $\displaystyle H_n = \left[ \begin{array}{cccc} 1 & \frac{1}{2} & \hdots & \frac{1}{n} \\ \frac{1}{2} & \frac{1}{3} & \hdots & \frac{1}{n+1} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{1}{n} & \frac{1}{n+1} & \hdots & \frac{1}{2n-1} \end{array} \right]$

and the corresponding Hankel determinants are

Image may be NSFW.
Clik here to view. $\displaystyle \frac{1}{h_n} = n! \prod_{i=1}^{2n-1} {i \choose \lfloor i/2 \rfloor}$ .

The corresponding state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ describes the uniform distribution on Image may be NSFW.
Clik here to view. [0, 1] .

Example. Consider the moment sequence Image may be NSFW.
Clik here to view. m_n = B_n , the Bell numbers. The corresponding Hankel determinants are

Image may be NSFW.
Clik here to view. $\displaystyle h_n = \prod_{i=1}^{n-1} i!$ .

The corresponding state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ describes the Poisson distribution with parameter Image may be NSFW.
Clik here to view..

Example. Consider the moment sequence with odd terms Image may be NSFW.
Clik here to view. $m_{2n-1} = 0$ and even terms

Image may be NSFW.
Clik here to view. $\displaystyle m_{2n} = 1 \cdot 3 \cdot ... \cdot (2n-1)$ .

The corresponding Hankel determinants are again

Image may be NSFW.
Clik here to view. $\displaystyle h_n = \prod_{i=1}^{n-1} i!$ .

The corresponding state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ describes the Gaussian distribution with mean Image may be NSFW.
Clik here to view. and variance Image may be NSFW.
Clik here to view.. As a corollary, the Hankel determinants of a moment sequence do not uniquely determine it.

Example. Consider the moment sequence Image may be NSFW.
Clik here to view. m_n = n! . The corresponding Hankel determinants are

Image may be NSFW.
Clik here to view. $\displaystyle h_n = \prod_{i=1}^{n-1} i!^2$ .

The corresponding state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ describes the exponential distribution with mean Image may be NSFW.
Clik here to view..

Example. Consider the moment sequence with odd terms Image may be NSFW.
Clik here to view. $m_{2n-1} = 0$ and even terms Image may be NSFW.
Clik here to view. $m_{2n} = C_n$ , the Catalan numbers. The corresponding Hankel determinants are

Image may be NSFW.
Clik here to view. $\displaystyle h_n = 1$ .

The corresponding state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ describes the Wigner semicircle distribution with radius Image may be NSFW.
Clik here to view.. The semicircle distribution is important in free probability, where it takes on a role analogous to the Gaussian distribution in a noncommutative version of the central limit theorem. It also appears in number theory as the Sato-Tate distribution, where it comes from the distribution of traces of a random element of Image may be NSFW.
Clik here to view. $\text{SU}(2)$ .

Example. Consider the moment sequence Image may be NSFW.
Clik here to view. m_n = C_n . The corresponding Hankel determinants are again

Image may be NSFW.
Clik here to view. $\displaystyle h_n = 1$ .

The corresponding state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ describes a random variable which is the square of a Wigner semicircularly distributed random variable.

Non-example. This example occurred a few years ago at the Secret Blogging Seminar. Consider the moment sequence with odd terms Image may be NSFW.
Clik here to view. $m_{2n-1} = 0$ and even terms

Image may be NSFW.
Clik here to view. $\displaystyle m_{2n} = \frac{3n+1}{n+1} {2n \choose n}$ .

The third Hankel determinant is

Image may be NSFW.
Clik here to view. $\displaystyle h_3 = \det \left[ \begin{array}{ccc} 1 & 0 & 4 \\ 0 & 4 & 0 \\ 4 & 0 & 14 \end{array} \right] = -8$ .

Hence this moment sequence does not define a state (and consequently cannot describe a random variable).

Orthogonal polynomials and Motzkin paths

Faithful states on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ are closely related to the classical theory of orthogonal polynomials. The starting point is that Image may be NSFW.
Clik here to view. $\mathbb{E}$ is a faithful state on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ if and only if Image may be NSFW.
Clik here to view. $\langle a, b \rangle = \mathbb{E}(a^{\dagger} b)$ defines an inner product on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ . Applying the Gram-Schmidt process with a slightly different normalization to the basis Image may be NSFW.
Clik here to view. $\{ 1, x, x^2, ... \}$ , we obtain a sequence Image may be NSFW.
Clik here to view. p_n of monic self-adjoint polynomials of degree Image may be NSFW.
Clik here to view., uniquely determined by the moment sequence Image may be NSFW.
Clik here to view. $m_n = \mathbb{E}(x^n)$ , such that

Image may be NSFW.
Clik here to view. $\displaystyle \langle p_n, p_m \rangle = \mathbb{E}(p_n p_m) = 0$

whenever Image may be NSFW.
Clik here to view. $n \neq m$ . (In addition, Image may be NSFW.
Clik here to view. $\mathbb{E}(p_n^2) > 0$ by faithfulness.) These are the orthogonal polynomials associated to Image may be NSFW.
Clik here to view. $\mathbb{E}$ (equivalently, to the moment sequence Image may be NSFW.
Clik here to view. m_n ).

Example. For a state describing a Wigner semicircular distribution with radius Image may be NSFW.
Clik here to view., the corresponding polynomials are (up to normalization) the Chebyshev polynomials of the second kind.

Example. For a state describing a Gaussian distribution with mean Image may be NSFW.
Clik here to view. and variance Image may be NSFW.
Clik here to view., the corresponding polynomials are the probabilist’s Hermite polynomials.

Example. For a state describing the uniform distribution on Image may be NSFW.
Clik here to view. [-1, 1] , the corresponding polynomials are (up to normalization) the Legendre polynomials.

Example. For a state describing the exponential distribution with mean Image may be NSFW.
Clik here to view., the corresponding polynomials are (up to normalization) the Laguerre polynomials.

The moments Image may be NSFW.
Clik here to view. $m_n = \mathbb{E}(x^n)$ can be evaluated in terms of the orthogonal polynomials Image may be NSFW.
Clik here to view. p_n as follows. First, by construction Image may be NSFW.
Clik here to view. p_k is orthogonal to all polynomials of degree at most Image may be NSFW.
Clik here to view. k-1 . Since Image may be NSFW.
Clik here to view. is self-adjoint with respect to the inner product above, Image may be NSFW.
Clik here to view. xp_k is orthogonal to all polynomials of degree at most Image may be NSFW.
Clik here to view. k-2 . It follows that the polynomials Image may be NSFW.
Clik here to view. $xp_k, p_{k+1}, p_k, p_{k-1}$ are orthogonal to all polynomials of degree at most Image may be NSFW.
Clik here to view. k-2 but have degree at most Image may be NSFW.
Clik here to view. k+1 , hence lie in a Image may be NSFW.
Clik here to view.-dimensional subspace of Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ . Moreover, by orthogonality Image may be NSFW.
Clik here to view. $p_{k+1}, p_k, p_{k-1}$ are linearly independent. Hence there exists a nontrivial linear dependence of the form

Image may be NSFW.
Clik here to view. $\displaystyle xp_k = p_{k+1} + a_k p_k + b_k p_{k-1}$

where the coefficient of Image may be NSFW.
Clik here to view. $p_{k+1}$ is determined by comparing leading coefficients. Thus the matrix of the linear operator given by multiplication by Image may be NSFW.
Clik here to view. with respect to the basis Image may be NSFW.
Clik here to view. $\{ p_0, p_1, p_2 ... \}$ is tridiagonal: it begins

Image may be NSFW.
Clik here to view. $\displaystyle \left[ \begin{array}{ccccc} a_0 & b_1 & 0 & 0 & \hdots \\ 1 & a_1 & b_2 & 0 & \hdots \\ 0 & 1 & a_2 & b_3 & \hdots \\ 0 & 0 & 1 & a_3 & \hdots \\ \vdots & \vdots & \vdots & \vdots & \ddots \end{array} \right]$ .

Corollary: Image may be NSFW.
Clik here to view. p_n is the characteristic polynomial of the matrix

Image may be NSFW.
Clik here to view. $\displaystyle J_n = \left[ \begin{array}{ccccc} a_0 & b_1 & 0 & \hdots & 0 \\ 1 & a_1 & b_2 & \hdots & 0 \\ 0 & 1 & a_2 & \hdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \hdots & a_{n-1}. \end{array} \right]$ .

Proof. Multiplication by Image may be NSFW.
Clik here to view. has characteristic polynomial Image may be NSFW.
Clik here to view. p_n on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]/p_n(x)$ , which has basis Image may be NSFW.
Clik here to view. $\{ p_0, ... p_{n-1} \}$ , and the above is the matrix by which Image may be NSFW.
Clik here to view. acts in this basis. Alternately, this can be proven by induction on Image may be NSFW.
Clik here to view. using the recurrence relation above. Image may be NSFW.
Clik here to view. $\Box$

It will be convenient to think of the above matrix as the weighted adjacency matrix of a weighted graph Image may be NSFW.
Clik here to view. with vertex set the non-negative integers Image may be NSFW.
Clik here to view. $\mathbb{Z}_{\ge 0}$ and, for every Image may be NSFW.
Clik here to view. $k \in \mathbb{Z}_{\ge 0}$ , three edges: an edge to Image may be NSFW.
Clik here to view. k+1 with weight Image may be NSFW.
Clik here to view., an edge to Image may be NSFW.
Clik here to view. with weight Image may be NSFW.
Clik here to view. a_k , and an edge to Image may be NSFW.
Clik here to view. k-1 with weight Image may be NSFW.
Clik here to view. b_k .

Image may be NSFW.
Clik here to view.

The Image may be NSFW.
Clik here to view. $n^{th}$ power of this matrix describes the sums of weights of paths of length Image may be NSFW.
Clik here to view. in this graph, and these weights describe the action of multiplication by Image may be NSFW.
Clik here to view. x^n with respect to the basis Image may be NSFW.
Clik here to view. $\{ p_0, p_1, p_2, ... \}$ . The corresponding paths (disregarding weights) are Motzkin paths, and they are counted by the Motzkin numbers Image may be NSFW.
Clik here to view. M_n .

In particular, Image may be NSFW.
Clik here to view. $\mathbb{E}(x^n) = \mathbb{E}(x^n p_0)$ is equal to the sum of the weights of all closed walks in Image may be NSFW.
Clik here to view. from Image may be NSFW.
Clik here to view. to itself of length Image may be NSFW.
Clik here to view.; this sum contains Image may be NSFW.
Clik here to view. M_n terms, one for each Motzkin path, and may be thought of as a weighted Motzkin number. The first few such sums are as follows:

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(1) = 1$

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(x) = a_0$

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(x^2) = a_0 \mathbb{E}(x) + b_1$

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(x^3) = a_0 \mathbb{E}(x^2) + a_1 b_1 + b_1 a_0$

Image may be NSFW.
Clik here to view. $\displaystyle \mathbb{E}(x^4) = a_0 \mathbb{E}(x^3) + b_1 a_0^2 + b_1^2 + a_1 b_1 a_0 + a_1^2 b_1 + b_2 b_1$ .

Example. For the Wigner semicircular distribution with Image may be NSFW.
Clik here to view. R = 2 , we have Image may be NSFW.
Clik here to view. a_n = 0, b_n = 1 for all Image may be NSFW.
Clik here to view.. The above expression for moments then reduces to a sum over Motzkin paths which never stay at a given vertex, hence over Dyck paths, which are counted by the Catalan numbers. There are no paths of odd length, and a path of length Image may be NSFW.
Clik here to view. has weight Image may be NSFW.
Clik here to view.. This gives Image may be NSFW.
Clik here to view. $\mathbb{E}(x^{2n-1}) = 0$ and Image may be NSFW.
Clik here to view. $\mathbb{E}(x^{2n}) = C_n$ as expected.

The combinatorial description of moments in terms of Motzkin paths leads to the following beautiful continued fraction expansion.

Theorem: We have an equality of formal power series

Image may be NSFW.
Clik here to view. $\displaystyle \sum_{n \ge 0} m_n t^n = \frac{1}{1 - a_0 t - \frac{b_1 t^2}{1 - a_1 t - \frac{b_2 t^2}{1 - a_2 t - ...}}}$ .

Proof. This is more difficult to formalize than to understand. Consider the weighted graph Image may be NSFW.
Clik here to view. described above. Let Image may be NSFW.
Clik here to view. $G_{\ge n}$ be the graph obtained from Image may be NSFW.
Clik here to view. by deleting the vertices Image may be NSFW.
Clik here to view. $\{ 0, 1, ... n-1 \}$ (and all corresponding edges), and let Image may be NSFW.
Clik here to view. W_n be the set of all paths from Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view. in Image may be NSFW.
Clik here to view. $G_{\ge n}$ . The combinatorial content of the above theorem is that a path in Image may be NSFW.
Clik here to view. W_n has a unique decomposition into a sequence of paths of the following two forms:

A loop Image may be NSFW.
Clik here to view. $n \to n$ (weight Image may be NSFW.
Clik here to view., length Image may be NSFW.
Clik here to view.), or
A step Image may be NSFW.
Clik here to view. $n \to n+1$ (weight Image may be NSFW.
Clik here to view., length Image may be NSFW.
Clik here to view.), a path in Image may be NSFW.
Clik here to view. $W_{n+1}$ , and a step Image may be NSFW.
Clik here to view. $n+1 \to n$ (weight Image may be NSFW.
Clik here to view. $b_{n+1}$ , length Image may be NSFW.
Clik here to view.).

Let Image may be NSFW.
Clik here to view. $\omega(W_n)$ denote the sum of all weights of all paths in Image may be NSFW.
Clik here to view. W_n , weighted in addition by Image may be NSFW.
Clik here to view. $t^{\text{length}}$ . Then Image may be NSFW.
Clik here to view. $\omega(W_0) = \sum m_n t^n$ , and the above argument shows that

Image may be NSFW.
Clik here to view. $\displaystyle \omega(W_n) = \frac{1}{1 - a_n t - b_{n+1} t^2 \omega(W_{n+1})}$ .

Applying this identity Image may be NSFW.
Clik here to view. times verifies the desired equality Image may be NSFW.
Clik here to view. $\bmod t^k$ , and taking Image may be NSFW.
Clik here to view. $k \to \infty$ gives the result by the universal property of formal power series. Image may be NSFW.
Clik here to view. $\Box$

A converse result

We saw above that we can associate to a faithful state Image may be NSFW.
Clik here to view. $\mathbb{E}$ on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ a sequence Image may be NSFW.
Clik here to view. p_n of monic polynomials of degree Image may be NSFW.
Clik here to view. such that Image may be NSFW.
Clik here to view. p_0 = 1 and

Image may be NSFW.
Clik here to view. $\displaystyle xp_n = p_{n+1} + a_n p_n + b_n p_{n-1}$

for some pair of sequences of real numbers Image may be NSFW.
Clik here to view. a_n, b_n uniquely determined by Image may be NSFW.
Clik here to view. $\mathbb{E}$ . This pair of sequences in turn uniquely determines the sequence of polynomials Image may be NSFW.
Clik here to view. p_n , hence uniquely determines the state Image may be NSFW.
Clik here to view. $\mathbb{E}$ via the conditions Image may be NSFW.
Clik here to view. $\mathbb{E}(p_0) = 1$ and Image may be NSFW.
Clik here to view. $\mathbb{E}(p_n) = 0, n \ge 1$ .

Given an arbitrary such pair of real sequences, it is natural to ask when the corresponding linear functional Image may be NSFW.
Clik here to view. $\mathbb{E}$ determines a faithful state.

Proposition: If Image may be NSFW.
Clik here to view. $n \neq m$ , then Image may be NSFW.
Clik here to view. $\mathbb{E}(p_n p_m) = 0$ , and Image may be NSFW.
Clik here to view. $\mathbb{E}(p_n^2) = b_1 b_2 ... b_n$ .

Proof. If Image may be NSFW.
Clik here to view. $n \neq m$ , assume WLOG that Image may be NSFW.
Clik here to view. n < m . Write Image may be NSFW.
Clik here to view. $p_n = \sum_{i=0}^n a_i x^i$ and apply the recurrence above to

Image may be NSFW.
Clik here to view. $\displaystyle p_n p_m = \sum_{i=0}^n a_i x^i p_m$

to write it in the basis Image may be NSFW.
Clik here to view. $\{ p_0, p_1, p_2, ... \}$ . Then Image may be NSFW.
Clik here to view. p_0 does not appear as a term. One way to see this is to use the combinatorial interpretation; the coefficients in Image may be NSFW.
Clik here to view. x^i p_m count paths on the weighted graph Image may be NSFW.
Clik here to view. starting at Image may be NSFW.
Clik here to view. of length Image may be NSFW.
Clik here to view. i < n , and such a path cannot return to the origin. Hence by construction Image may be NSFW.
Clik here to view. $\mathbb{E}(p_n p_m) = 0$ .

If Image may be NSFW.
Clik here to view. n = m , then applying the recurrence to Image may be NSFW.
Clik here to view. p_n p_n we see that the only contribution to the coefficient of Image may be NSFW.
Clik here to view. p_0 comes from the unique path from Image may be NSFW.
Clik here to view. to Image may be NSFW.
Clik here to view. of length Image may be NSFW.
Clik here to view., which has weight Image may be NSFW.
Clik here to view. b_1 b_2 ... b_n as desired. Image may be NSFW.
Clik here to view. $\Box$

Corollary (Favard’s theorem): The linear functional on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ associated to a pair of real sequences Image may be NSFW.
Clik here to view. a_n, b_n as above is a faithful state if and only if Image may be NSFW.
Clik here to view. b_n > 0 for all Image may be NSFW.
Clik here to view..

Proof. Since the Image may be NSFW.
Clik here to view. p_n are orthogonal, Image may be NSFW.
Clik here to view. $\mathbb{E}$ is faithful if and only if Image may be NSFW.
Clik here to view. $\mathbb{E}(p_n^2) > 0$ for all Image may be NSFW.
Clik here to view.. Image may be NSFW.
Clik here to view. $\Box$

This gives us a method to construct faithful states on Image may be NSFW.
Clik here to view. $\mathbb{C}[x]$ without computing Hankel determinants: we can instead write down a sequence Image may be NSFW.
Clik here to view. a_n of real numbers and another sequence Image may be NSFW.
Clik here to view. b_n of positive real numbers, then compute the corresponding orthogonal polynomials. The corresponding moment sequence can be computed using Motzkin paths or using the continued fraction. Alternatively, given a moment sequence which we suspect determines a faithful state, we can write down what we suspect the corresponding orthogonal polynomials are and compute the sequences Image may be NSFW.
Clik here to view. a_n and Image may be NSFW.
Clik here to view. b_n to verify that Image may be NSFW.
Clik here to view. b_n > 0 for all Image may be NSFW.
Clik here to view..

Example. Taking Image may be NSFW.
Clik here to view. a_n = 0, b_n = 1 for all Image may be NSFW.
Clik here to view. gives the Wigner semicircular distribution with Image may be NSFW.
Clik here to view. R = 2 .

Computing Hankel determinants

We now give the promised result which explains the Hankel determinants given without proof above.

Theorem: With notation as above, we have

Image may be NSFW.
Clik here to view. $\displaystyle h_n = \prod_{i=1}^{n-1} b_1 b_2 ... b_i = \prod_{i=1}^{n-1} b_i^{n-i}$

Proof 1. First, observe that the change of basis matrix from the basis Image may be NSFW.
Clik here to view. $\{ 1, x, x^2, ... \}$ to the basis Image may be NSFW.
Clik here to view. $\{ p_0, p_1, p_2, ... \}$ is by construction triangular with Image may be NSFW.
Clik here to view.s on the diagonal. Such a change of basis changes the Hankel matrices by a congruence Image may be NSFW.
Clik here to view. $H_n \mapsto P_n^T H_n P_n$ where Image may be NSFW.
Clik here to view. P_n has determinant Image may be NSFW.
Clik here to view., and consequently does not affect the value of the Hankel determinants. We can therefore compute the Hankel determinants with respect to the basis of orthogonal polynomials. By construction, the Hankel matrices with respect to Image may be NSFW.
Clik here to view. $\{ p_0, p_1, p_2, ... \}$ are diagonal:

Image may be NSFW.
Clik here to view. $\displaystyle H_n = \left[ \begin{array}{ccccc} \mathbb{E}(p_0^2) & 0 & 0 & \hdots & 0 \\ 0 & \mathbb{E}(p_1^2) & 0 & \hdots & 0 \\ 0 & 0 & \mathbb{E}(p_2^2)& \hdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \hdots & \mathbb{E}(p_{n-1}^2) \end{array} \right]$ .

It follows that

Image may be NSFW.
Clik here to view. $\displaystyle h_n = \prod_{i=0}^{n-1} \mathbb{E}(p_i^2) = \prod_{i=1}^{n-1} b_1 b_2 ... b_i$

as desired. Image may be NSFW.
Clik here to view. $\Box$

Proof 2. We will give a combinatorial proof using the Lindström-Gessel-Viennot lemma. Consider the following locally finite directed acyclic graph Image may be NSFW.
Clik here to view. $\tilde{G}$ : the vertices are the set of pairs Image may be NSFW.
Clik here to view. $(x, y) \in \mathbb{Z}^2$ with Image may be NSFW.
Clik here to view. $y \ge 0$ . The edges take the following forms:

edges Image may be NSFW.
Clik here to view. $(x, i) \to (x + 1, i)$ with weights Image may be NSFW.
Clik here to view.,
edges Image may be NSFW.
Clik here to view. $(x, y) \to (x + 1, y + 1)$ with weight Image may be NSFW.
Clik here to view., and
edges Image may be NSFW.
Clik here to view. $(x, i + 1) \to (x + 1, i)$ with weights Image may be NSFW.
Clik here to view. $b_{i+1}$ .

For a fixed positive integer Image may be NSFW.
Clik here to view., take the Image may be NSFW.
Clik here to view. sources in the statement of the LGV lemma to be the vertices Image may be NSFW.
Clik here to view. (0, 0), (-1, 0), ... (-(n-1), 0) and take the Image may be NSFW.
Clik here to view. sinks to be the vertices Image may be NSFW.
Clik here to view. (0, 0), (1, 0), ... (n-1, 0) . Then the paths from Image may be NSFW.
Clik here to view. (-i, 0) to Image may be NSFW.
Clik here to view. (j, 0) may be identified with closed walks of length Image may be NSFW.
Clik here to view. i + j on the weighted Motzkin graph Image may be NSFW.
Clik here to view. (in the sense that there is a weight-preserving bijection between them), so the matrix appearing in the statement of the LGV lemma is precisely the Hankel matrix Image may be NSFW.
Clik here to view. H_n (with respect to the basis Image may be NSFW.
Clik here to view. $\{ 1, x, x^2, ... \}$ ).

Image may be NSFW.
Clik here to view.

On the other hand, there is a unique non-intersecting Image may be NSFW.
Clik here to view.-path in Image may be NSFW.
Clik here to view. $\tilde{G}$ : the source Image may be NSFW.
Clik here to view. (-i, 0) is taken to the sink Image may be NSFW.
Clik here to view. (i, 0) by Image may be NSFW.
Clik here to view. diagonal steps up and to the right, then Image may be NSFW.
Clik here to view. diagonal steps down and to the right, and this is the unique possibility by induction on Image may be NSFW.
Clik here to view.. This path has weight Image may be NSFW.
Clik here to view. b_1 b_2 ... b_i , from which the conclusion follows by the LGV lemma. Image may be NSFW.
Clik here to view. $\Box$

Example. For the Wigner semicircular distribution with Image may be NSFW.
Clik here to view. R = 2 , we know that Image may be NSFW.
Clik here to view. a_n = 0, b_n = 1 for all Image may be NSFW.
Clik here to view., which gives Image may be NSFW.
Clik here to view. h_n = 1 for all Image may be NSFW.
Clik here to view. as desired.

Moments, Hankel determinants, orthogonal polynomials, Motzkin paths, and continued fractions

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