Proof step e-27282dcb1ed61456

preserver_image affine, shift 0 → concludes about basis-map coefficients of Riordan array (1, 3+x). 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\":\"-4\"}"
}

Premises

• Riordan array (1, 3+x). (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4/9, 0]

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "32/9" }, "upper": { "closed": true, "value": "4" } } }
• { "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 78dbce0e151fb92885e03e7641bce3ee01e92951fbbc66572db2656e25b05cc0, 19,515 bytes (3,819 compressed); download JSON

Cited by

• n-943ef8677f31ba50 (real_rooted)
• n-943ef8677f31ba50 (one_signed_coefficients)
• n-943ef8677f31ba50 (interlacing)