Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver reversal, shift 0

Concludes

reversal of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).

Premises

Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
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-d5ee6f5e013acb0f · 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 430fb4d8e00dc204c12f61db808a9aa35b0f679e4831553dadee01c56b92d0ea, 16,900 bytes (3,267 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-940e4d08e5526f20 (real_rooted)
n-940e4d08e5526f20 (one_signed_coefficients)
n-940e4d08e5526f20 (hadamard_kernel)