(1) p_{n+1}(x)=(x-b_n)*p_n(x)-\lambda_n*p_{n-1}(x), n>0, p_0(x)=1, p_{-1}(x)=0.
Assume that \lambda_n>0, and that L is the positive definite linear functional for which p_n is orthogonal, L(1)=1.
Suppose that S is another linear functional such that S(p_k*p_l)=0 for n>=k>l>=0, S(1)=1. Show that S has the same j th moments as L, for j at most 2n-1.
2. Explicitly renormalize the polynomials p_n(x) to p_n*(x) in problem 1 so that the eigenvalues and eigenvectors of the related real symmetric tridiagonal n by n matrix are explicitly given by the zeros x_{n,k} of p_n(x) and the vectors (p_0*(x_{n,k}),...,p_{n-1}*(x_{n,k})). Find these results.
Next, using orthogonality of the eigenvectors, rederive the Gaussian quadrature formula
L_n(p)=\sum_{k=1}^n \Lambda_{nk} p(x_{nk}).
Then use problem 1 to conclude that L_n and L have the identical moments up to 2n-1.
3. Let \Delta(x)=\prod_{k>j}(x_k-x_j) be the Vandermonde determinant in n variables x_1,...,x_n. Let w(x)=x^a(1-x)^b on [-1,1], where a,b>-1. Evaluate
(a) \int_0^1 ...\int_0^1 \Delta(x)^2 w(x_1)w(x_2)...w(x_n) dx_1...dx_n
(b) \int_0^1 ...\int_0^1 \Delta(x)^2 w(x_1)w(x_2)...w(x_n) e_k(x_1,...,x_n) dx_1...dx_n, where e_k is the elementary symmetric function of degree k.
4. The (monic) Stieltjes-Wigert polynomials p_n(x) may be defined by
p_n(x)=q^(-n^2-n/2)*(-1)^n*\sum_{j=0}^n [n choose j]_q*q^(j^2)*(-q^(1/2)*x)^j.
(a) Prove that p_n satisfies (1) with
b_n=-(q-(1+q)q^(-n))*q^(-n-3/2), \lambda_n=(1-q^n)*q^(-4n)
(b) Show that p_n is orthogonal with respect to the moment sequence L(x^n)=q^(-n^2/2-n).
(c) Show that a representing measure for L is a multiple of
k*x^(-k^2*log(x))dx, on [0,\infty), where q=e^(-1/(2k^2)).