Proof step

Veronese sections of Pólya frequency sequences

Veronese coefficient section r=2, j=1, shift 0

Concludes

Binomial coefficients C(n,k) with n-k even, read by rows.

Premises

veronese parent of Binomial coefficients C(n,k) with n-k even, read by rows.
sequence: Every row P_n, n >= 0, 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-1d1ac0038b6609d9 · JSON

Operation

Veronese coefficient section

{
  "index_shift": 0,
  "modulus": 2,
  "operation": "veronese_section",
  "residue": 1
}

Claims

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

Certificate

veronese_section, SHA-256 c133b089a5b526eed9207db7e0da9eec23439cfc9ccb9b5224020152fba8cddd, 4,531 bytes (1,293 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