Proof step e-c61963b254c15dd4

preserver_image affine, shift 0 → concludes about basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments. 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

• Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "-1" } } }
• { "claim": "interlacing", "from_index": 1, "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 6ab1e51b285f2352394f49c093faf5f3e5c077d1cda0558d9a6eb6bb6cc25962, 23,498 bytes (5,872 compressed); download JSON

Cited by

• n-359eaa7b22014e19 (hadamard_kernel)
• n-359eaa7b22014e19 (interlacing)
• n-359eaa7b22014e19 (real_rooted)
• n-359eaa7b22014e19 (one_signed_coefficients)