Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

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

Premises

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 (-5, -1)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-6582417c7c9b5874 · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "lower": {
      "closed": false,
      "value": "-6"
    },
    "upper": {
      "closed": false,
      "value": "-2"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 1,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 b86a1997aee92c5feb7cdf2e900c5aea8731468b6c9ce53081e396fca185d817, 18,407 bytes (3,660 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-30cb3cad216ed189 (interlacing)
n-30cb3cad216ed189 (one_signed_coefficients)
n-30cb3cad216ed189 (hadamard_kernel)
n-30cb3cad216ed189 (real_rooted)