Proof step e-c5713bb0c1f908c6

preserver_image mobius, shift 0 → concludes about mobius_x_over_x_plus_one of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k. 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\":1,\"first_degree\":0,\"step\":1}}"
}

Premises

• Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-4, 0)

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": true, "value": "-1" }, "upper": { "closed": false, "value": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 3fdfb7ef95e3a045cfcdb7664f6ef4cb3499d24b74a8d4c015f6acdd9bf78eb9, 17,097 bytes (3,259 compressed); download JSON

Cited by

• n-ac316f680714c5b7 (real_rooted)
• n-ac316f680714c5b7 (one_signed_coefficients)
• n-ac316f680714c5b7 (hadamard_kernel)