Chebyshev polynomials of the second kind

TheoremCoefficients of the Chebyshev polynomials of the second kind#✓fully proved

Let \(U_n \in \mathbb{Z}[x]\) denote the \(n\)-th Chebyshev polynomial of the second kind, and let \(k \le n\) be natural numbers. If \(n\) and \(k\) have the same parity, the coefficient of \(x^k\) in \(U_n\) is \((-1)^{(n-k)/2} \sum_{i=0}^{k} \binom{(n+k)/2}{i} \binom{(n+k)/2 - i}{(n-k)/2}\); otherwise it is \(0\).

theorem chebyshevU_coeff_eq_binomial_sum
    (n k : ℕ) (hk : k ≤ n) :
    (Polynomial.Chebyshev.U ℤ (n : ℤ)).coeff k =
      if n % 2 = k % 2 then
        (-1 : ℤ) ^ ((n - k) / 2) *
          ∑ i ∈ Finset.range (k + 1),
            ((Nat.choose ((n + k) / 2) i : ℕ) : ℤ) *
              ((Nat.choose ((n + k) / 2 - i) ((n - k) / 2) : ℕ) : ℤ)
      else 0

Sources

• Milan Janjić, "On a Class of Polynomials with Integer Coefficients", Journal of Integer Sequences 11 (2008), Article 08.5.2, Corollary lines 441-446, https://cs.uwaterloo.ca/journals/JIS/VOL11/Janjic/janjic19.tex.

DefinitionModified Chebyshev base polynomials of the second kind#✓fully proved

For a commutative ring \(R\) and \(r, s \in R\), the polynomials \(P_n \in R[x]\), \(n \ge 0\), are defined by \(P_0 = 1\), \(P_1 = x - r\) and \(P_{n+2} = (x - r) P_{n+1} - s P_n\) for every \(n \ge 0\).

noncomputable def P (r s : R) : ℕ → Polynomial R
  | 0 => 1
  | 1 => X - C r
  | n + 2 => (X - C r) * P r s (n + 1) - C s * P r s n

Used by 2

Used byGeneralized Chebyshev polynomials of the second kind · Three-term recurrence for generalized Chebyshev polynomials

DefinitionGeneralized Chebyshev polynomials of the second kind#✓fully proved

For a commutative ring \(R\) and \(r, s, \lambda, \mu \in R\), let \(P_n \in R[x]\) be the modified Chebyshev base polynomials with parameters \(r, s\). The polynomials \(Q_n \in R[x]\) are defined by \(Q_0 = 1\), \(Q_1 = P_1 - \lambda P_0\) and \(Q_{n+2} = P_{n+2} - \lambda P_{n+1} - \mu P_n\) for every \(n \ge 0\).

noncomputable def Q (r s lam mu : R) (n : ℕ) : Polynomial R :=
  match n with
  | 0 => 1
  | 1 => P r s 1 - C lam * P r s 0
  | n + 2 => P r s (n + 2) - C lam * P r s (n + 1) - C mu * P r s n

Uses 1 · Used by 1

Statement usesModified Chebyshev base polynomials of the second kind

Used byThree-term recurrence for generalized Chebyshev polynomials

TheoremThree-term recurrence for generalized Chebyshev polynomials#✓fully proved

For a commutative ring \(R\), all \(r, s, \lambda, \mu \in R\) and every \(n \ge 0\), the generalized Chebyshev polynomials of the second kind \(Q_n \in R[x]\) with parameters \(r, s, \lambda, \mu\) satisfy \(Q_{n+3} = (x - r) Q_{n+2} - s Q_{n+1}\).

No hypothesis on \(\mu\) is needed because the recurrence starts at \(Q_3\); the analogous identity \(Q_2 = (x - r) Q_1 - s Q_0\) holds only when \(\mu = 0\).

theorem Q_recurrence (r s lam mu : R) (n : ℕ) :
    Q r s lam mu (n + 3)
      = (X - C r) * Q r s lam mu (n + 2) - C s * Q r s lam mu (n + 1)

Uses 2

Statement usesGeneralized Chebyshev polynomials of the second kind

Proof usesModified Chebyshev base polynomials of the second kind

4 fully proved

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