Cornell University Math REU

• Background
• Monomials
• Recurrence Relation
• SUMS Poster
• Home

Recurrence Relation

Denote by {p_k | k=0,…,∞} the orthogonal polynomials obtained from {P_j3 | j=0,…,∞} by the Gram-Schmidt Process. The p_k are antisymmetric orthogonal polynomials.

Define the polynomials ρ_j = P_j1 + R¹(P_j1) + R²(P_j1), where R¹ is rotation once CCW, and R² is rotation twice CCW. Therefore, the ρ_j are fully symmetric polynomials.

Then we denote by {s_k | k=0,…,∞} the orthogonal polynomials obtained from {ρ_j | j=0,…,∞} by the Gram-Schmidt Process. The s_k are fully symmetric orthogonal polynomials.

And Green's function is defined so that if u solves Δu = f, then:

u(x) = − ∫ G(x,y) f(y) dμ(y)

We also define the inner product to be:

< f, g > = ∫ f(x) g(x) dμ(x)

Then we have the following main result.

Theorem 1

Let {p_k | k=0,…,∞} be the orthogonal polynomials defined above. Let f₀(x)=0 and for k≥0, let f_k+1 be the polynomial of degree k+1 with leading coefficient 1 given by

f_k+1(x):= − ∫ G(x,y) p_k(y) dμ(y)

Set p_-1(x):=0, and p₀(x)=P₀₃(x). Then, for each k≥ 0

p_k+1(x) = f_k+1(x) − b_k p_k(x) − c_k p_k-1(x)

where

b_k = d_k² < f_k+1,p_k > ,
c_k= d_k-1² / d_k² = ||p_k||² / ||p_k-1||²

Consequently,

d_k^-2 = ||p_k||² = d₀^-2 c₁ c₂ c₃ … c_k-1 c_k

---

Then, by normalizing the p_k, we get the orthonormal polynomials Q_k:

Q_k = d_k p_k = p_k / ||p_k||

By a similar procedure, we can calulate the s_k and their orthonormal counterparts S_k.

Furthermore, we note here that the S_k and the rotations of the Q_k form a tight frame for their span in H^j, k≤j. As a result, we define the fully orthonormal polynomial φ_k by:

φ_k = √(2/3) Q_k + √(1/3) S_k

Note that φ_k and its rotations for k≤j form an orthonormal basis for H^j