Proof step e-6ddcf5c6f38774e7

obreschkoff_combination → concludes about obreschkoff_combination of Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2. JSON

Theorem

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

Operation

Obreschkoff combination Pₙ + Pₙ₊₁

{
  "operation": "obreschkoff_combination"
}

Premises

• Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2. (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
• Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r = 2. (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 c8694901a5e1ee741bc89ad0a4b8be8ee6c3815dcf5790116fa2864f6678eab3, 5,921 bytes (1,603 compressed); download JSON

Cited by

• n-954f0c79bf5b2ac0 (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.