Proof step e-b21ed9aa36fa01d5

preserver_image affine, shift 0 → concludes about Exponential Riordan array [x*exp(x),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

• Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated … (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": { "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 39ac11f8c6d99d8f426894cfb7a74922582e017b64cb7609ad73e6e02a821e2e, 9,005 bytes (2,652 compressed); download JSON

Cited by

• n-308cf60736924b4c (one_signed_coefficients)
• n-308cf60736924b4c (hadamard_kernel)
• n-308cf60736924b4c (real_rooted)