Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

shift_x_plus_one of reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).

Premises

reversal of Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, 0)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-0fa740c32486add6 · 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": "-5"
    },
    "upper": {
      "closed": false,
      "value": "-1"
    }
  }
}

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

Certificate

preserver_image, SHA-256 0a7291b3cc84c1be7905171e2e470e9b196a5ce5eb6921d5262b3368a29ca964, 19,023 bytes (3,505 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-02870b15db72fb9a (real_rooted)
n-02870b15db72fb9a (one_signed_coefficients)
n-02870b15db72fb9a (hadamard_kernel)
n-02870b15db72fb9a (interlacing)