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 shift_x_plus_one of chebyshev_u

Premises

shift_x_plus_one of chebyshev_u
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)

Theorem

polytool ?

Lean

RealRooted route fTransform RealRooted.splits_fPolynomial

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-2e59a4494301abb0 · 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": false,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 fea632369243cb0a48ce6f56e4230ac457043812118f39f2ce743700c06f1a20, 17,850 bytes (3,198 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-12d187bc73368c94 (real_rooted)
n-12d187bc73368c94 (one_signed_coefficients)
n-12d187bc73368c94 (hadamard_kernel)