Proof step e-b8a3e660dc9132dc

preserver_image mobius, shift 0 → concludes about mobius_x_over_x_plus_one 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): route fTransform RealRooted.splits_fPolynomial

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\":0,\"first_degree\":0,\"step\":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": { "lower": { "closed": true, "value": "-1" }, "upper": { "closed": true, "value": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 7c917b01eeeab01719f7b749b657ba4c972416cd20569fe89e69c9d26bd44377, 22,088 bytes (5,460 compressed); download JSON

Cited by

• n-9af5c0b0557bf778 (real_rooted)
• n-9af5c0b0557bf778 (one_signed_coefficients)
• n-9af5c0b0557bf778 (hadamard_kernel)