Proof step e-5342624c2722e8d8

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle read by rows of coefficients of Chebyshev's U(n,x) polynomials (exponents in increasing order). JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

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

Premises

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

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "-2" }, "upper": { "closed": false, "value": "0" } } }
• { "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 2bafd0b330c5a973950215909e69e63674ff1db403c75df293d9b9434b244b31, 18,121 bytes (3,160 compressed); download JSON

Cited by

• n-b1a7e7698038fd9e (real_rooted)
• n-b1a7e7698038fd9e (one_signed_coefficients)
• n-b1a7e7698038fd9e (hadamard_kernel)
• n-b1a7e7698038fd9e (interlacing)