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 Riordan array ((1+x)^3,x).

Premises

shift_x_plus_one of Riordan array ((1+x)^3,x).
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-a0dc9830dd4c34e0 · 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\":3,\"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 b1c1544316d2b7ecac118d7edcb57f0440ebbc90eb30b1a6054f266780848071, 8,223 bytes (2,340 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-ad0bc0d16494c943 (real_rooted)
n-ad0bc0d16494c943 (one_signed_coefficients)
n-ad0bc0d16494c943 (hadamard_kernel)