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 reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).

Premises

reversal of Riordan array (1/(1-2*x), x*(1+x)/(1-2*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-bd27bbb2ad5e3b5d · 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 8cda4f105e22dcd50a7323455d9553d1c74cffdd825863d05058b3c6a345253b, 28,139 bytes (4,877 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-b48b62a5f27e6dcd (real_rooted)
n-b48b62a5f27e6dcd (one_signed_coefficients)
n-b48b62a5f27e6dcd (hadamard_kernel)