Proof step e-07f8adddfb9eb09c

obreschkoff_combination → concludes about obreschkoff_combination of Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <=… 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, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n). (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, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >= 1, 1 <= m <= n). (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 140c343d32e525b06d380d567b311998ad47f1632feaba9bb1fadea6b595ad3f, 6,033 bytes (1,657 compressed); download JSON

Cited by

• n-e92c310aea8cdd6a (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.