Proof step e-5d14c25d8e006de4

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\":\"-1\"}"
}

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": "5/9" }, "upper": { "closed": true, "value": "1" } } }
• { "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 35eb7e8e7dee1a9c553ad8f33a813a2d33bc6a0966d24ddd450da3614d5762be, 19,510 bytes (3,815 compressed); download JSON

Cited by

• n-b0cf60708a253825 (real_rooted)
• n-b0cf60708a253825 (one_signed_coefficients)
• n-b0cf60708a253825 (interlacing)