Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

Triangle of coefficients of Legendre polynomials 2^n P_n (x).

Premises

Expansion of 1/sqrt(1-4*x*y-4*x^2*y).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-43f4e7bf47d18868 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 0948d12322a247534fd3357f7d27c358af3dafd78ced467796897537c2e684e6, 10,949 bytes (2,861 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-b2f14d9f44a39df2 (hadamard_kernel)
n-b2f14d9f44a39df2 (real_rooted)
n-b2f14d9f44a39df2 (one_signed_coefficients)