Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver mobius, shift 0

Concludes

mobius_x_over_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 route fTransform RealRooted.splits_fPolynomial

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-ff65129705903856 · 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\":1,\"first_degree\":0,\"step\":1}}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "lower": {
      "closed": true,
      "value": "-1"
    },
    "upper": {
      "closed": false,
      "value": "0"
    }
  }
}

Certificate

preserver_image, SHA-256 9ec0a2f2e5cd9a0715f73062a8e456d3c2df7d0a2e84dfd7f9359190de59d6e8, 17,897 bytes (3,232 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-de6b5f28ca03e00c (real_rooted)
n-de6b5f28ca03e00c (one_signed_coefficients)
n-de6b5f28ca03e00c (hadamard_kernel)