Proof step e-3e84156cf8108ace

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients. 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

• Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients. (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" } } }
• { "claim": "interlacing", "from_index": 1, "kind": "weak", "orientation": "unordered" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 4d9fb46b9d5bbf30fad4c298c039c16a8de664582ad4b5525a496d40dd2855ba, 11,812 bytes (2,986 compressed); download JSON

Cited by

• n-f339d17ab0e2d2a9 (real_rooted)
• n-f339d17ab0e2d2a9 (one_signed_coefficients)
• n-f339d17ab0e2d2a9 (hadamard_kernel)
• n-f339d17ab0e2d2a9 (interlacing)