Proof step e-c8c4101b2f8f8c16

preserver_image affine, shift 0 → concludes about basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n. 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\":\"0\"}"
}

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": { "lower": { "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 3c74c5ad69fc1899f4d918b494ecb6cf61ff531fc217a2ee43f6954ee266777c, 4,786 bytes (1,493 compressed); download JSON

Cited by

• n-50902364601606aa (real_rooted)
• n-50902364601606aa (one_signed_coefficients)