Proof step e-51a320c1f4e00092

preserver_image affine, shift 0 → concludes about shift_x_plus_one of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0 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 Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0 (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 fc419bf81d03d7dcedfcaee5cff057aaf2b0159f564718a4d92b0f35f7a129ff, 4,783 bytes (1,483 compressed); download JSON

Cited by

• n-f8c9dbb903f44fdc (real_rooted)
• n-f8c9dbb903f44fdc (one_signed_coefficients)
• n-f8c9dbb903f44fdc (hadamard_kernel)