Proof step e-aefe61e44b642324

preserver_image affine, shift 0 → concludes about shift_x_plus_one of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. 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

• basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. (sequence)
  uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Claims

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

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 7969d508a4f46610f4ea23f9c6daad419451d798f5c6da350b5d332a4bafd25e, 4,835 bytes (1,503 compressed); download JSON

Cited by

• n-a6f555b0e6271a68 (real_rooted)
• n-a6f555b0e6271a68 (one_signed_coefficients)
• n-a6f555b0e6271a68 (hadamard_kernel)