Proof step e-2126fae20f2e5116

preserver_image hadamard_multiplier, shift 0 → concludes about Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2). JSON

Theorem

Pólya–Schur theorem on multiplier sequences [SP14; Bra15, Thm 40] polytool ?

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"hadamard_multiplier\",\"factor\":{\"terms\":[{\"kind\":\"binomial\",\"index_offset\":0,\"coefficient_offset\":0}]}}"
}

Premises

• Irregular triangle read by rows: T(n,k) = binomial(n-k, k), n >= 0, 0 <= k <= floor(n/2); or, coefficients of (one version of) Fibonacci po… (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 b8c5f55917df763144235a87895da38a6d07b64a2d57068c6c6da3edb1820de5, 10,932 bytes (2,863 compressed); download JSON

Cited by

• n-8bfdd58279be42d7 (real_rooted)
• n-8bfdd58279be42d7 (one_signed_coefficients)
• n-8bfdd58279be42d7 (hadamard_kernel)

References

1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.