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_t

Premises

shift_x_plus_one of chebyshev_t
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-3d5368d261b111d1 · 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 b47f04da77e4e15d82f68836aa7e1f1f4391db689fa48b32109287640f523794, 19,093 bytes (3,314 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-e9f3812507785414 (real_rooted)
n-e9f3812507785414 (one_signed_coefficients)
n-e9f3812507785414 (hadamard_kernel)