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 Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].

Premises

Sheffer triangle (exp(x), exp(3*x) - 1). Named S2[3,1].
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-0b30d1ed25425a81 · 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 3e9d188828c1365b162a97c051b153eab0ccadd99dd057b44b3b648e4ec08cbf, 10,864 bytes (2,731 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-4e3b9877e6d1f6d9 (one_signed_coefficients)
n-4e3b9877e6d1f6d9 (hadamard_kernel)
n-4e3b9877e6d1f6d9 (real_rooted)