Proof step e-039745f356dca7e2

obreschkoff_combination → concludes about obreschkoff_combination of Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k). JSON

Theorem

Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

Operation

Obreschkoff combination Pₙ + Pₙ₊₁

{
  "operation": "obreschkoff_combination"
}

Premises

• Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
• Triangle read by rows, T(n, k) = binomial(n, k)*binomial(n + 2, k). (sequence)
  uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1

Claims

• { "claim": "real_rooted", "from_index": 0, "region": {} }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: obreschkoff_combination, SHA-256 28ddaddb1ea2be9a20359679e1143ad6fbe297eb9c15c72ffe14495b931e66a6, 16,584 bytes (3,952 compressed); download JSON

Cited by

• n-137a24c9ba6939cb (real_rooted)

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
2. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.