Proof step e-c36241f3b645d7b5

preserver_image reversal, shift 0 → concludes about Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)). JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): route reversed RealRooted.X_pow_mul_reverse_eq_zero_or_splits, RealRooted.reflect_eq_zero_or_splits

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"
}

Premises

• Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments. (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": "0" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 706675af6eb6e7ebcc4bee3d8038d1e8dbb84c29c672fc24e4a1e53f18a510c8, 22,237 bytes (5,542 compressed); download JSON

Cited by

• n-d6b7687ba0ff59dd (one_signed_coefficients)
• n-d6b7687ba0ff59dd (hadamard_kernel)
• n-d6b7687ba0ff59dd (real_rooted)