Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

shift_x_plus_one of shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).

Premises

shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order).
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-2, 0)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-3b84462f43b7ccde · 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": 1,
  "region": {
    "lower": {
      "closed": false,
      "value": "-3"
    },
    "upper": {
      "closed": false,
      "value": "-1"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 a866f08f101cc3f1f9ff23a6c4946aa1101236fe8e46d275e008cdebd5321034, 18,843 bytes (3,457 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-5c55f31aa1da8314 (real_rooted)
n-5c55f31aa1da8314 (one_signed_coefficients)
n-5c55f31aa1da8314 (hadamard_kernel)
n-5c55f31aa1da8314 (interlacing)