Proof step

Laguerre’s theorem on differential operators

Image under a real-rootedness preserver derivative polynomial, shift 0

Concludes

Triangle read by rows: 2-Stirling numbers of the second kind.

Premises

Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
sequence: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]

Theorem

[SP14] polytool ?

Lean

RealRooted RealRooted.splits_add_C_mul_derivative_all, RealRooted.splits_C_mul_add_derivative, Polynomial.Splits.derivative

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-6ee0d58dba9c37ea · JSON

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"
}

Claims

{
  "claim": "real_rooted",
  "from_index": 1,
  "region": {
    "upper": {
      "closed": true,
      "value": "0"
    }
  }
}

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

Certificate

preserver_image, SHA-256 bca68ceb9ccc4544163ac517b33298f87ee98699e9eb36164306d0a848e80d9c, 13,274 bytes (3,112 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-6aaec68d653cbdb0 (one_signed_coefficients)
n-6aaec68d653cbdb0 (hadamard_kernel)
n-6aaec68d653cbdb0 (real_rooted)
n-6aaec68d653cbdb0 (interlacing)

References

[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.