Proof step

Veronese sections of Pólya frequency sequences

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

Concludes

Premises

Exponential Riordan array [x*exp(x),x].
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[ASW52; Edr53; AW24] polytool ?

Checked

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Technical details

Step id

e-cf0ef1200ee3dfa0 · 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"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "previous_interlaces_next"
}

Certificate

veronese_section, SHA-256 04f33c1dfcdccf65e0487e84f62020b6b7cbb453980903784a90d00e41e22326, 11,255 bytes (3,012 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