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+5) polynomials (exponents of x in increasing order).

Premises

Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-6, -2)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-e1a6f38ce117cf24 · 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": "-7"
    },
    "upper": {
      "closed": false,
      "value": "-3"
    }
  }
}

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

Certificate

preserver_image, SHA-256 87d2d0f1cb7df9a7ab60926087852fafddc87229b9045cdf3cae14ea01948542, 18,485 bytes (3,682 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-c2b7c74f8491f01b (interlacing)
n-c2b7c74f8491f01b (one_signed_coefficients)
n-c2b7c74f8491f01b (hadamard_kernel)
n-c2b7c74f8491f01b (real_rooted)