Proof step e-0787c477c17aca9a

obreschkoff_combination → concludes about obreschkoff_combination of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k). JSON

Theorem

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

Operation

Obreschkoff combination Pₙ + Pₙ₊₁

{
  "operation": "obreschkoff_combination"
}

Premises

• Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
• Twice Pascal's triangle A007318: T(n,k) = 2*C(n,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 e02cd713f73c0a8517fab695efce4cc901e15fbff0a9b3d5baa7805c6b3805fd, 8,502 bytes (2,205 compressed); download JSON

Cited by

• n-2ab5cee8137b2478 (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.