Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

mobius_x_over_x_plus_one of A Jacobsthal-Pascal triangle.

Premises

A Jacobsthal-Pascal triangle.
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

polytool ?

Lean

RealRooted route fTransform RealRooted.splits_fPolynomial

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-33ed1f696f0d9ad5 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"mobius\",\"a\":\"1\",\"b\":\"0\",\"c\":\"1\",\"e\":\"1\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "lower": {
      "closed": true,
      "value": "-1"
    },
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 fdc8e6093f66e0a8fc6704b3d90a59bdf1f55e430045258ede518b863a075ca0, 9,392 bytes (2,742 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-4a6b9237eddcc625 (real_rooted)
n-4a6b9237eddcc625 (one_signed_coefficients)
n-4a6b9237eddcc625 (hadamard_kernel)