Proof step

Veronese sections of Pólya frequency sequences

Veronese coefficient section r=3, j=1, shift 0

Concludes

Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).

Premises

veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[ASW52; Edr53; AW24] polytool ?

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-1e55567759b4d30f · JSON

Operation

Veronese coefficient section

{
  "index_shift": 0,
  "modulus": 3,
  "operation": "veronese_section",
  "residue": 1
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

Certificate

veronese_section, SHA-256 88ef48c3882eeebc4925b3e86790bfdd862cede81b0a06a7feb81606b02900a6, 4,969 bytes (1,308 compressed); download

polytool replayed the certificate when the step was written.

Cited by

References

[ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
[AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
[Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3