Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

reversal of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

Premises

reversal of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).
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-fd03420edaf2e6be · 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 45efb111130685fe651d6a178054e11bc07c22767a44d98593a8c5f1c0e73552, 27,115 bytes (4,673 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-8ac1223942f7b8c5 (one_signed_coefficients)
n-8ac1223942f7b8c5 (hadamard_kernel)
n-8ac1223942f7b8c5 (real_rooted)