shiftedLegendre — Mathlib

English

A restatement of the contraction equality of degrees in a simplified form

Русский

Упрощённое утверждение о равенстве степеней конракций

LaTeX

$$$\\forall q\\; [hq : NeZero\\ q]\\ [CharP\\ F\\ q]\\ (g\\ g' : F[X]) (m\\ m' : \\mathbb{N}),\\; expand\\ F (q^m) g = expand\\ F (q^{m'}) g' \\Rightarrow g.natDegree = g'.natDegree$$$

Lean4

/-- `shiftedLegendre n` is an integer polynomial for each `n : ℕ`, defined by:
`Pₙ(x) = ∑ k ∈ Finset.range (n + 1), (-1)ᵏ * choose n k * choose (n + k) n * xᵏ`
These polynomials appear in combinatorics and the theory of orthogonal polynomials. -/
noncomputable def shiftedLegendre (n : ℕ) : ℤ[X] :=
  ∑ k ∈ Finset.range (n + 1), C ((-1 : ℤ) ^ k * n.choose k * (n + k).choose n) * X ^ k

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