Proof step

Constant-current basis map

Basis map theorem for the sequence's own rule

Concludes: Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.
Premises: none: a direct theorem check
Theorem: [Bra15, Lemma 8.5 of arXiv:1410.6601] polytool RealRooted orderTwoBasis_pf_equivalences RealRooted basisTransform_orderTwoBasis_preserves_bounded_pf_iff ?
Checked: polytool replayed · Lean not attempted

Technical details

Step id

e-2052c3f3275d98f4 · JSON

Operation

Basis map theorem for the sequence's own rule

{
  "operation": "basis_map_theorem"
}

Claims

{
  "claim": "basis_map_preserver",
  "input_interval": "(-inf, 0]",
  "output_interval": "(-inf, 0]",
  "statement": "every PF input has a real-rooted image with every root <= -c/b = 0"
}

{
  "claim": "basis_map_preserver",
  "input_interval": "(-inf, 0]",
  "output_interval": "(-inf, 0]",
  "statement": "every PF input has a PF image (zero allowed)"
}

Certificate

basis_map_theorem, SHA-256 46b55f7e6d81849679eb7d83fe8d3f47dac292942122470cfe54f87ae12228b6, 3,083 bytes (1,010 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-fe6a74fcc4c3def7 (basis_map_preserver)

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10